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In $3x^2$, the number multiplying $x^2$ is 3."},{"text":"2","correct":false,"explanation":"The 2 is the index, not the coefficient. The coefficient is the number multiplying $x^2$, which is 3."},{"text":"$$x^2$","correct":false,"explanation":"The variable part has been selected instead of its numerical factor. In $3x^2$, the coefficient is 3, while $x^2$ is the powered variable term."},{"text":"1","correct":false,"explanation":"A coefficient of 1 would usually be implied when no number is written. Here $x^2$ is explicitly multiplied by 3, so the coefficient is 3."}]},{"id":"86616799-35ee-4376-801c-5609fb17378f","cardType":"basic","points":1,"front":"State the index law for negative powers, $a^{-n}$.","back":"$$a^{-n} = \\frac{1}{a^n}$","mcqOptions":[{"text":"$$a^{-n} = \\frac{1}{a^n}$","correct":true,"explanation":"A negative index means take the reciprocal of the corresponding positive power, so $a^{-n}$ becomes $\\frac{1}{a^n}$."},{"text":"$$a^{-n} = -a^n$","correct":false,"explanation":"The negative sign in an index does not make the value negative; it indicates a reciprocal. Therefore $a^{-n} = \\frac{1}{a^n}$."},{"text":"$$a^{-n} = a^n$","correct":false,"explanation":"Changing a negative index to a positive one without taking a reciprocal loses the essential index-law rule. The correct form is $\\frac{1}{a^n}$."},{"text":"$$a^{-n} = \\frac{1}{a^{-n}}$","correct":false,"explanation":"The reciprocal must contain the positive power $a^n$, not the original negative power. Thus the correct result is $a^{-n} = \\frac{1}{a^n}$."}]},{"id":"7413e83b-0965-43cc-8b01-ca54636bbb91","cardType":"cloze","points":null,"front":"To simplify a fractional power like $a^{\\frac{m}{n}}$, you take the [ ] and the $m$th power.","back":"To simplify a fractional power like $a^{\\frac{m}{n}}$, you take the $n$th root and the $m$th power.","mcqOptions":[{"text":"$$n$th root","correct":true,"explanation":"The denominator of the fractional index gives the root, so $a^{\\frac{m}{n}}$ involves taking the $n$th root before applying the $m$th power."},{"text":"$$m$th root","correct":false,"explanation":"The numerator $m$ determines the power, not the root. The denominator $n$ gives the root, so the correct term is the $n$th root."},{"text":"$$n$th power","correct":false,"explanation":"The denominator of a fractional index determines the root rather than the power. The $n$th root must be taken before the $m$th power."},{"text":"$$m$th power","correct":false,"explanation":"The $m$th power is the second operation in this expression, because the numerator gives the power. The first operation is taking the $n$th root."}]},{"id":"5d031d17-8086-4fa0-934d-6a165b2e40dd","cardType":"cloze","points":null,"front":"To simplify a fractional power like $a^{\\frac{m}{n}}$, you take the $n$th root and the [ ].","back":"To simplify a fractional power like $a^{\\frac{m}{n}}$, you take the $n$th root and the $m$th power.","mcqOptions":[{"text":"$$m$th power","correct":true,"explanation":"The numerator of the fractional index gives the power, so after taking the $n$th root, you apply the $m$th power."},{"text":"$$n$th power","correct":false,"explanation":"The denominator $n$ determines the root, not the power. The numerator $m$ gives the required power, so the correct term is the $m$th power."},{"text":"$$m$th root","correct":false,"explanation":"The numerator $m$ determines the power rather than the root. The root is set by the denominator $n$, followed by the $m$th power."},{"text":"$$n$th root","correct":false,"explanation":"The $n$th root is the first operation and has already been stated. The remaining operation is the $m$th power, determined by the numerator."}]},{"id":"18052ef9-c9c5-4cf0-a764-ce1569a4e7a0","cardType":"basic","points":1,"front":"How do you rewrite $27^{2x-1}$ as a power of 3?","back":"$$(3^3)^{2x-1} = 3^{3(2x-1)}$","mcqOptions":[{"text":"$$(3^3)^{2x-1} = 3^{3(2x-1)}$","correct":true,"explanation":"Since $27=3^3$, replace the base with $3^3$. The power-of-a-power law then multiplies the indices, giving $3^{3(2x-1)}$."},{"text":"$$(3^3)^{2x-1} = 3^{2x-1}","correct":false,"explanation":"This keeps the new base but omits the outer index multiplying the inner index. Using $(a^m)^n=a^{mn}$ gives $3^{3(2x-1)}$."},{"text":"$$(3^3)^{2x-1} = 3^{3(2x+1)}","correct":false,"explanation":"The sign has been changed from subtraction to addition inside the exponent. Rewriting the base does not alter $2x-1$, so the exponent is $3(2x-1)$."},{"text":"$$(3^2)^{2x-1} = 3^{2(2x-1)}$","correct":false,"explanation":"The base conversion is incorrect because $27=3^3$, not $3^2$. Therefore the correct rewrite is $(3^3)^{2x-1}=3^{3(2x-1)}$."}]},{"id":"4c0c1cb3-46d3-44df-9b42-c991eb10fc04","cardType":"cloze","points":null,"front":"When simplifying $(a^m)^n$, you must [ ] the indices together.","back":"When simplifying $(a^m)^n$, you must multiply the indices together.","mcqOptions":[{"text":"multiply","correct":true,"explanation":"The power-of-a-power law states $(a^m)^n=a^{mn}$, so the indices are multiplied together when simplifying $(a^m)^n$."},{"text":"add","correct":false,"explanation":"Adding indices applies when multiplying powers with the same base, not when raising a power to another power. Here the indices must be multiplied."},{"text":"divide","correct":false,"explanation":"Dividing indices is not the rule for a power raised to another power. The correct law is $(a^m)^n=a^{mn}$, which multiplies them."},{"text":"subtract","correct":false,"explanation":"Subtracting indices is associated with dividing powers of the same base, not a power raised to a power. In this expression, the indices are multiplied."}]},{"id":"b9426767-c3c9-4f0c-af0d-3cdcefff5699","cardType":"basic","points":1,"front":"What is the common base for an equation containing $8^x$ and $4^x$?","back":"Base 2 ($2^3$ and $2^2$)","mcqOptions":[{"text":"Base 2 ($2^3$ and $2^2$)","correct":true,"explanation":"Both 8 and 4 can be written as powers of 2: $8=2^3$ and $4=2^2$. This gives the expressions a common base for comparison or solving."},{"text":"Base 3 ($3^2$ and $3^1$)","correct":false,"explanation":"Although 8 and 4 are the given bases, neither equals the stated powers of 3. They are correctly expressed as $2^3$ and $2^2$, so the common base is 2."},{"text":"Base 4 ($4^2$ and $4^1$)","correct":false,"explanation":"This would make the first expression $16$, not 8. The integer powers needed are $8=2^3$ and $4=2^2$, so the common base is 2."},{"text":"Base 8 ($8^1$ and $8^2$)","correct":false,"explanation":"The second expression would become $64$ rather than 4. Rewriting both original bases with integer powers uses base 2: $8=2^3$ and $4=2^2$."}]},{"id":"0572816f-8da2-4de5-a6ce-56f6e12293e5","cardType":"basic","points":2,"front":"What substitution converts $4^x - 10(2^x) + 16 = 0$ into a quadratic equation?","back":"Let $y = 2^x$ (giving $y^2 - 10y + 16 = 0$)","mcqOptions":[{"text":"Let $y = 2^x$ (giving $y^2 - 10y + 16 = 0$)","correct":true,"explanation":"Because $4^x=(2^2)^x=(2^x)^2$, setting $y=2^x$ changes the equation into $y^2-10y+16=0$, a quadratic in $y$."},{"text":"Let $y = 4^x$ (giving $y^2 - 10y + 16 = 0$)","correct":false,"explanation":"With $y=4^x$, the term $2^x$ is $\\sqrt{y}$, so the equation does not become the stated quadratic. Setting $y=2^x$ makes $4^x=y^2$."},{"text":"Let $y = 2x$ (giving $y^2 - 10y + 16 = 0$)","correct":false,"explanation":"The substitution must involve the exponential expression $2^x$, not the product $2x$. Setting $y=2^x$ converts $4^x$ into $y^2$ and gives the quadratic."},{"text":"Let $y = 2^{x+1}$ (giving $y^2 - 10y + 16 = 0$)","correct":false,"explanation":"This substitution introduces an extra factor of 2, since $2^{x+1}=2\\cdot2^x$. The direct substitution $y=2^x$ is needed to obtain $y^2-10y+16=0$."}]},{"id":"b049afc5-ae7e-48ae-8732-e0ab2e8815ee","cardType":"basic","points":2,"front":"If solving an exponential equation via substitution $y = a^x$, why must a result of $y = -2$ be rejected?","back":"An exponential term $a^x$ (where $a > 0$) is always positive.","mcqOptions":[{"text":"An exponential term $a^x$ (where $a > 0$) is always positive.","correct":true,"explanation":"For every real value of x, a^x is positive when a > 0. Therefore, y = -2 cannot correspond to any value of x and must be rejected."},{"text":"An exponential term $a^x$ (where $a > 0$) is always greater than 1.","correct":false,"explanation":"The base being positive does not restrict y to values above 1; for example, 2^x can equal 1/2. The actual restriction is that a^x is always positive, so y = -2 is impossible."},{"text":"An exponential term $a^x$ (where $a > 0$) is never equal to zero.","correct":false,"explanation":"Exponential expressions can equal zero only as a limiting value, never for a finite real x. The relevant reason for rejecting y = -2 is that a^x is always positive."},{"text":"An exponential term $a^x$ (where $a > 0$) is always an integer.","correct":false,"explanation":"A positive base can produce negative exponents, such as 2^{-1} = 1/2. The decisive restriction is positivity: a^x cannot be negative, so y = -2 must be rejected."}]},{"id":"66b1590c-c01d-468d-a046-1121fbd01b9b","cardType":"cloze","points":null,"front":"The expression $2^{2x}$ is equivalent to the square of $\\text{[...]}$.","back":"The expression $2^{2x}$ is equivalent to the square of $\\boxed{2^x}$.","mcqOptions":[{"text":"$$2^x$","correct":true,"explanation":"Using the index law $(a^m)^n = a^{mn}$, squaring $2^x$ gives $(2^x)^2 = 2^{2x}$. Thus the missing term is $2^x$."},{"text":"$$2^{x+1}$","correct":false,"explanation":"Squaring $2^{x+1}$ gives $2^{2x+2}$, not $2^{2x}$. The exponent must be x before squaring, so the correct term is $2^x$."},{"text":"$$2^{x-1}$","correct":false,"explanation":"Squaring $2^{x-1}$ gives $2^{2x-2}$, which is smaller by a factor of 4. The required term is $2^x$, whose square has exponent 2x."},{"text":"$$2^{2x}$","correct":false,"explanation":"Squaring $2^{2x}$ would produce $2^{4x}$, doubling the exponent again. The expression already has exponent 2x, so its square root in this form is $2^x$."}]},{"id":"474b205e-e333-44f8-9c13-3d3fc88bb1d1","cardType":"basic","points":2,"front":"What are the two main steps to factorise $6x - 24x^3$ completely?","back":"

Factorise out the highest common factor ($6x$)
Apply difference of two squares to $(1 - 4x^2)$

","mcqOptions":[{"text":"

Factorise out the highest common factor ($6x$)
Apply difference of two squares to $(1 - 4x^2)$

","correct":true,"explanation":"Both terms contain 6x, so removing it gives 6x(1 - 4x^2). The bracket is then a difference of squares: 1^2 - (2x)^2."},{"text":"

Factorise out the highest common factor ($6x^2$)
Apply difference of two squares to $(1 - 4x)$

","correct":false,"explanation":"The common factor is 6x, not 6x^2, because the first term contains only one x. After removing it, the remaining bracket is handled using a difference of two squares."},{"text":"

Factorise out the highest common factor ($-6x$)
Apply sum of two squares to $(1 + 4x^2)$

","correct":false,"explanation":"Taking out -6x changes the bracket to 4x^2 - 1, which can also be factorised, but this option gives the stated bracket incorrectly and does not follow the requested main steps."},{"text":"

Factorise out the highest common factor ($24x^3$)
Apply difference of two squares to $(1 - 6x)$

","correct":false,"explanation":"The expression has a common factor of 6x, not 24x^3. Removing the entire second term as a factor is impossible because 6x is not divisible by 24x^3."}]},{"id":"873ffc67-9c21-41cc-9837-65174bb91e8e","cardType":"cloze","points":null,"front":"A negative fractional index like $\\left(\\frac{a}{b}\\right)^{-\\frac{1}{2}}$ indicates you should first take the [ ] of the fraction.","back":"A negative fractional index like $\\left(\\frac{a}{b}\\right)^{-\\frac{1}{2}}$ indicates you should first take the reciprocal of the fraction.","mcqOptions":[{"text":"reciprocal","correct":true,"explanation":"A negative index means invert the base before applying the fractional index. Therefore, the first operation is to take the reciprocal of the fraction."},{"text":"square root","correct":false,"explanation":"The square root corresponds to the denominator 2 in the fractional index, but it is not the first operation caused by the negative sign. The fraction must first be reciprocated."},{"text":"negative","correct":false,"explanation":"The negative sign is part of the index and does not name an operation to perform on the fraction. Its effect is inversion, so the reciprocal is taken first."},{"text":"conjugate","correct":false,"explanation":"A conjugate changes the sign between terms such as a + √b and a - √b. It is unrelated to the negative index here, which requires taking the reciprocal first."}]},{"id":"0ff5e075-1af0-44e5-8715-d3e6b2d7917b","cardType":"basic","points":1,"front":"Expand and simplify $(x + 1)^2$.","back":"$$x^2 + 2x + 1$","mcqOptions":[{"text":"$$x^2 + 2x + 1$","correct":true,"explanation":"Applying $(a+b)^2 = a^2 + 2ab + b^2$ with a = x and b = 1 gives $x^2 + 2x + 1$."},{"text":"$$x^2 + x + 1$","correct":false,"explanation":"This result omits the middle term from the expansion. The two cross-products x·1 and 1·x combine to give 2x, so the correct result includes that term."},{"text":"$$x^2 + 2x + 2$","correct":false,"explanation":"The middle term has the correct coefficient, but the constant term should be 1^2 = 1, not 2. The complete expansion is $x^2 + 2x + 1$."},{"text":"$$x^2 - 2x + 1$","correct":false,"explanation":"The sign of the middle term is positive because both terms in the bracket are positive. A negative middle term would arise from $(x - 1)^2$, not this expression."}]},{"id":"16b0ee32-8fa4-4ddb-8683-c49f71bd4127","cardType":"basic","points":1,"front":"What is the conjugate of $a + \\sqrt{b}$ used in rationalising denominators?","back":"$$a - \\sqrt{b}$","mcqOptions":[{"text":"$$a - \\sqrt{b}$","correct":true,"explanation":"The conjugate changes the sign between the two terms while leaving both terms otherwise unchanged. Multiplying by it creates $(a + \\sqrt{b})(a - \\sqrt{b}) = a^2 - b$."},{"text":"$$-a + \\sqrt{b}$","correct":false,"explanation":"Changing the sign of a only does not form the conjugate. The radical sign must change, giving $a - \\sqrt{b}$ for the conjugate of the original expression."},{"text":"$$a + \\sqrt{b}$","correct":false,"explanation":"Keeping the plus sign leaves the expression unchanged rather than forming a conjugate. The conjugate must have a minus between a and the square root."},{"text":"$$-a - \\sqrt{b}$","correct":false,"explanation":"Changing both signs gives the negative of the original expression, not its conjugate. Only the sign between the rational and surd terms changes."}]},{"id":"a7b32917-b555-462a-94ec-7b1ff5db18d4","cardType":"cloze","points":null,"front":"Multiplying a denominator by its conjugate rationalises it by creating a [ ].","back":"Multiplying a denominator by its conjugate rationalises it by creating a difference of two squares.","mcqOptions":[{"text":"difference of two squares","correct":true,"explanation":"For conjugate factors, $(a + b)(a - b) = a^2 - b^2$, so the cross terms cancel and the radical denominator becomes rational. This is the difference of two squares."},{"text":"sum of two squares","correct":false,"explanation":"A sum of two squares does not result from multiplying conjugates; the cross terms cancel with opposite signs. The resulting structure is a difference of two squares."},{"text":"perfect square","correct":false,"explanation":"The product of conjugates is not generally a perfect square because the terms have different signs. It forms a difference of squares after the cross terms cancel."},{"text":"common factor","correct":false,"explanation":"Conjugate multiplication removes the mixed terms rather than producing a common factor. The algebraic identity involved is the difference of two squares."}]},{"id":"5bc0cba0-ddef-4581-a5d0-682bbac17998","cardType":"basic","points":2,"front":"How do you solve for $x$ in an equation like $\\frac{2 + x}{x} = \\sqrt{5}$?","back":"

Multiply by $x$ to clear the fraction
Collect $x$ terms on one side and factorise out $x$
Divide by the coefficient bracket

","mcqOptions":[{"text":"

Multiply by $x$ to clear the fraction
Collect $x$ terms on one side and factorise out $x$
Divide by the coefficient bracket

","correct":true,"explanation":"Multiplying by x gives $2 + x = x\\sqrt{5}$. Collecting x terms produces $2 = x(\\sqrt{5} - 1)$, then division by the coefficient bracket isolates x."},{"text":"

Add $x$ to both sides to clear the fraction
Collect constant terms on one side and factorise out $x$
Multiply by the coefficient bracket

","correct":false,"explanation":"Adding x to both sides does not remove the denominator. Multiplication by x is required first, after which x terms are collected and the resulting coefficient bracket is divided out."},{"text":"

Divide by $x$ to clear the fraction
Collect $x$ terms on one side and expand the coefficient bracket
Subtract the coefficient bracket

","correct":false,"explanation":"Dividing by x again makes the denominator more complicated and does not clear it. The equation should be multiplied by x before collecting and factoring the x terms."},{"text":"

Multiply by $\\sqrt{5}$ to clear the fraction
Collect $x$ terms on one side and factorise out $x$
Divide by $x$

","correct":false,"explanation":"The numerator cannot be treated as a separate factor before clearing the fraction. Multiply by x, collect the x terms, factorise them, and divide by their coefficient bracket."}]},{"id":"c9a4da40-7bb2-42d2-9546-a552d168c1e0","cardType":"cloze","points":null,"front":"Strictly speaking, $\\sqrt{x^2} = \\text{[...]}$.","back":"Strictly speaking, $\\sqrt{x^2} = \\boxed{|x|}$.","mcqOptions":[{"text":"$$|x|$","correct":true,"explanation":"The principal square root is always non-negative, so squaring and then taking the square root gives the magnitude of $x$, which is $|x|$."},{"text":"$$x$","correct":false,"explanation":"Writing $x$ ignores negative values: if $x=-3$, then $\\sqrt{x^2}=3$, not $-3$. The expression therefore simplifies to $|x|$."},{"text":"$$-x$","correct":false,"explanation":"The principal square root cannot be negative, so $-x$ fails whenever $x$ is positive. The result valid for every real $x$ is $|x|$."},{"text":"$$x^2$","correct":false,"explanation":"This omits the effect of taking the square root and leaves the squared quantity unchanged. Since the principal root is non-negative, $\\sqrt{x^2}=|x|$."}]},{"id":"d24f73dc-8692-4604-a874-2cda0e084734","cardType":"basic","points":1,"front":"How do you rewrite $a^m \\times a^n$ as a single power?","back":"$$a^{m+n}$","mcqOptions":[{"text":"$$a^{m+n}$","correct":true,"explanation":"When powers with the same base are multiplied, their exponents are added. Therefore $a^m \\times a^n$ combines to give $a^{m+n}$."},{"text":"$$a^{mn}$","correct":false,"explanation":"Multiplying the exponents is a common confusion with raising a power to another power. When multiplying powers with the same base, add the exponents to obtain $a^{m+n}$."},{"text":"$$a^{m-n}$","correct":false,"explanation":"Subtracting exponents applies when dividing powers with the same base, not multiplying them. Here the exponents must be added, giving $a^{m+n}$."},{"text":"$$(2a)^{m+n}$","correct":false,"explanation":"Adding the bases is not part of the index law, because the common base remains unchanged. Multiplication of like bases gives $a^{m+n}$."}]},{"id":"7c59ace0-136b-4830-9c5e-81ed444a8c79","cardType":"basic","points":1,"front":"What is the result of $\\frac{a^m}{a^n}$?","back":"$$a^{m-n}$","mcqOptions":[{"text":"$$a^{m-n}$","correct":true,"explanation":"When powers with the same non-zero base are divided, the denominator’s exponent is subtracted from the numerator’s exponent, producing $a^{m-n}$."},{"text":"$$a^{m+n}$","correct":false,"explanation":"Adding exponents is the rule for multiplying powers with the same base. Division requires subtracting the denominator’s exponent, so the result is $a^{m-n}$."},{"text":"$$a^{\\frac{m}{n}}$","correct":false,"explanation":"Dividing the exponents is not an index law for quotients of like bases. Instead, subtract the denominator’s exponent from the numerator’s to get $a^{m-n}$."},{"text":"$$a^{n-m}$","correct":false,"explanation":"Reversing the subtraction gives the reciprocal of the required result. Since the numerator contains $a^m$, the correct exponent is $m-n$, giving $a^{m-n}$."}]},{"id":"29aabfb9-0e8c-4bcf-91c2-3165549d3784","cardType":"basic","points":1,"front":"Define like terms in an algebraic expression.","back":"Terms that share exactly the same variable part (e.g., $5xy$ and $-2xy$)","mcqOptions":[{"text":"Terms that share exactly the same variable part (e.g., $5xy$ and $-2xy$)","correct":true,"explanation":"Like terms must have identical variable factors, including the same variables raised to the same powers. Their numerical coefficients may differ, as with $5xy$ and $-2xy$."},{"text":"Terms that share at least one variable part (e.g., $5xy$ and $-2xz$)","correct":false,"explanation":"Sharing only one variable is insufficient: for example, $3xy$ and $4xz$ cannot be combined. Like terms must have exactly the same variable part."},{"text":"Terms that share exactly the same coefficient part (e.g., $5xy$ and $5x^2$)","correct":false,"explanation":"Coefficients do not need to match for terms to be like; they are the values added or subtracted. The variable parts must instead be exactly identical."},{"text":"Terms that share the same variables in any powers (e.g., $5xy$ and $-2x^2y$)","correct":false,"explanation":"Having the same variables is not enough if their powers differ: $5xy$ and $-2x^2y$ are unlike terms. The entire variable part must match exactly."}]}]}],"examCountdown":{"examDate":"2027-05-13","paperLabel":"Paper 1"},"heading":"Algebraic Expressions","title":"Algebraic Expressions Flashcards","topbar":{"backHref":"/as-level/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"6ec4b7bb-12a0-4d28-b1a4-f9e82baa4442","walled":"$L39"}],"$L3a","$L3b"]