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3 is the base
4 is the index (or power)

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

3 is the base
4 is the index (or power)

","correct":true,"explanation":"The base is the number being repeatedly multiplied, while the index tells you how many copies of the base are multiplied together."},{"text":"

3 is the index
4 is the base (or power)

","correct":false,"explanation":"This reverses the two terms. In $3^4$, 3 is the base being multiplied and 4 is the index, which specifies four copies of 3."},{"text":"

3 is the factor
4 is the multiplier (or total)

","correct":false,"explanation":"Factor and multiplier describe multiplication more generally, not the positions in power notation. In $3^4$, 3 is the base and 4 is the index."},{"text":"

3 is the exponent
4 is the coefficient (or factor)

","correct":false,"explanation":"This confuses power notation with algebraic coefficients. In $3^4$, 3 is the base, while 4 is the index or exponent."}]},{"id":"045436dc-428d-cbbc-bae8-9187d4c4647e","cardType":"cloze","points":null,"front":"In the expression $3^4$, the index 4 tells you to multiply [ ] copies of the base together.","back":"In the expression $3^4$, the index 4 tells you to multiply four copies of the base together.","mcqOptions":[{"text":"four","correct":true,"explanation":"An index states how many copies of the base are multiplied together. Therefore, $3^4$ means $3 \\times 3 \\times 3 \\times 3$."},{"text":"three","correct":false,"explanation":"This mistakes the base for the number of copies. The index is 4, so four copies of the base 3 are multiplied together."},{"text":"two","correct":false,"explanation":"Two copies would represent $3^2$, not $3^4$. The index 4 means that four copies of 3 must be multiplied together."},{"text":"five","correct":false,"explanation":"This incorrectly counts one more copy than the index states. Since the index is 4, exactly four copies of the base are multiplied."}]},{"id":"33bc2675-d1ba-e89e-12ec-77ea5eb7746e","cardType":"basic","points":1,"front":"Evaluate $3^3$.","back":"27","mcqOptions":[{"text":"27","correct":true,"explanation":"The index 3 means multiply three copies of the base: $3^3 = 3 \\times 3 \\times 3 = 27$."},{"text":"9","correct":false,"explanation":"This evaluates $3^2$ by multiplying only two copies of 3. For $3^3$, three copies are required, giving $3 \\times 3 \\times 3 = 27$."},{"text":"6","correct":false,"explanation":"This comes from multiplying the base by the index, $3 \\times 3$. An index represents repeated multiplication, so $3^3 = 27$."},{"text":"81","correct":false,"explanation":"This uses four copies of 3 and therefore evaluates $3^4$. The index in $3^3$ requires only three copies, giving 27."}]},{"id":"79bb9150-de56-8145-a5e8-e58be34c37f2","cardType":"basic","points":2,"front":"Why is $5^2$ not equal to 10?","back":"$$5^2$ means 5 multiplied by itself ($5 \\times 5$), not multiplied by 2.","mcqOptions":[{"text":"$$5^2$ means 5 multiplied by itself ($5 \\times 5$), not multiplied by 2.","correct":true,"explanation":"The index 2 tells you to use two copies of the base 5. Therefore, $5^2 = 5 \\times 5 = 25$, rather than 10."},{"text":"$$5^2$ means 5 multiplied by 2 ($5 \\times 2$), so it is equal to 10.","correct":false,"explanation":"This treats the index as an ordinary multiplier. The index gives the number of copies of the base, so $5^2 = 5 \\times 5 = 25$."},{"text":"$$5^2$ means 5 added to itself ($5 + 5$), not multiplied by itself.","correct":false,"explanation":"This confuses repeated multiplication with repeated addition. Squaring 5 requires multiplying two copies of 5, so $5^2 = 25$, not 10."},{"text":"$$5^2$ means 2 multiplied by itself ($2 \\times 2$), with 5 as the index.","correct":false,"explanation":"This reverses the base and index. In $5^2$, 5 is the base and 2 is the index, so the calculation is $5 \\times 5 = 25$."}]},{"id":"b530ee90-d2cf-e52a-112a-7f2b1f07b3d4","cardType":"basic","points":1,"front":"Write the repeated multiplication $6 \\times 6 \\times 6 \\times 6$ in power notation.","back":"$6^4$","mcqOptions":[{"text":"$6^4$","correct":true,"explanation":"There are four identical factors of 6. Power notation writes the repeated factor as the base and the number of copies as the index, giving $6^4$."},{"text":"$4^6$","correct":false,"explanation":"This reverses the base and index. The repeated factor is 6 and it appears four times, so the correct power notation is $6^4$."},{"text":"$6^3$","correct":false,"explanation":"This misses one of the four factors and represents only $6 \\times 6 \\times 6$. Four copies of 6 must be written as $6^4$."},{"text":"$24^1$","correct":false,"explanation":"This multiplies 6 by the number of factors instead of representing repeated multiplication. Four factors of 6 are written as $6^4$, not as $24^1$."}]},{"id":"dd3194f4-2970-57c8-8855-e6b37b59e46c","cardType":"cloze","points":null,"front":"A [ ] is the answer you get when a number is multiplied by itself.","back":"A square number is the answer you get when a number is multiplied by itself.","mcqOptions":[{"text":"square number","correct":true,"explanation":"A square number is produced by multiplying an integer by itself. For example, $7 \\times 7 = 49$, so 49 is a square number."},{"text":"cube number","correct":false,"explanation":"A cube number comes from multiplying three copies of a number, not two. Multiplying a number by itself once produces a square number."},{"text":"square root","correct":false,"explanation":"A square root is a number that produces a given value when squared. The answer produced by multiplying a number by itself is a square number."},{"text":"prime number","correct":false,"explanation":"A prime number has exactly two positive factors, which is unrelated to this definition. Multiplying a number by itself produces a square number."}]},{"id":"0da9eaf7-0a4a-f434-1bce-77f5ef55469d","cardType":"basic","points":1,"front":"What are the first five square numbers?","back":"1, 4, 9, 16, 25","mcqOptions":[{"text":"1, 4, 9, 16, 25","correct":true,"explanation":"These are obtained by squaring the first five positive integers: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$ and $5^2 = 25$."},{"text":"1, 2, 4, 8, 16","correct":false,"explanation":"These are powers of 2 rather than consecutive square numbers. Squaring the first five positive integers gives 1, 4, 9, 16 and 25."},{"text":"1, 3, 9, 27, 81","correct":false,"explanation":"These are powers of 3, not the results of squaring consecutive positive integers. The first five square numbers are 1, 4, 9, 16 and 25."},{"text":"2, 4, 6, 8, 10","correct":false,"explanation":"These are the first five positive even numbers, not square numbers. Squaring 1, 2, 3, 4 and 5 gives 1, 4, 9, 16 and 25."}]},{"id":"3b1e7b02-48ee-efc2-9ab8-1c84ac50b13e","cardType":"basic","points":1,"front":"Is the square of an even number (like $6^2$) even or odd?","back":"Even (e.g. 36)","mcqOptions":[{"text":"Even (e.g. 36)","correct":true,"explanation":"An even number contains a factor of 2, so multiplying it by itself still gives a multiple of 2. For example, $6^2 = 36$."},{"text":"Odd (e.g. 49)","correct":false,"explanation":"The example 49 is the square of the odd number 7, not an even number. Squaring any even number produces an even result, such as $6^2 = 36$."},{"text":"It can be either (e.g. 36 or 49)","correct":false,"explanation":"This confuses squares of even numbers with squares in general. While 49 is an odd square, the square of an even number is always even."},{"text":"Neither (e.g. 36)","correct":false,"explanation":"Every integer is either even or odd, and 36 is even because it is divisible by 2. The square of an even number is always even."}]},{"id":"eba0ddf0-0fd3-0f9f-2437-9fc5ab85f37f","cardType":"basic","points":1,"front":"What question should you ask yourself to find $\\sqrt{100}$?","back":"What number multiplied by itself gives 100?","mcqOptions":[{"text":"What number multiplied by itself gives 100?","correct":true,"explanation":"A square root identifies the number that produces the original value when multiplied by itself. Since $10 \\times 10 = 100$, asking this question gives $\\sqrt{100}=10$."},{"text":"What number multiplied by 2 gives 100?","correct":false,"explanation":"Multiplying by 2 finds half of 100, which is 50, rather than its square root. Ask what number multiplied by itself gives 100; that number is 10."},{"text":"What number divided by itself gives 100?","correct":false,"explanation":"Any non-zero number divided by itself equals 1, not 100. To find a square root, ask what number multiplied by itself gives 100; the answer is 10."},{"text":"What number added to itself gives 100?","correct":false,"explanation":"Adding a number to itself finds half of 100, which is 50. Square roots involve multiplication, so ask what number multiplied by itself gives 100; it is 10."}]},{"id":"12ee7c2d-1ae4-4777-f422-dacf4cd5ed76","cardType":"basic","points":1,"front":"Evaluate $\\sqrt{64}$.","back":"8","mcqOptions":[{"text":"8","correct":true,"explanation":"The square root of 64 is the positive number that gives 64 when squared. Since $8 \\times 8 = 64$, $\\sqrt{64}=8$."},{"text":"6","correct":false,"explanation":"This may come from confusing 64 with 36. Since $6^2=36$, not 64, the correct value is 8 because $8^2=64$."},{"text":"16","correct":false,"explanation":"This may result from dividing 64 by 4 rather than taking its square root. The correct answer is 8 because $8 \\times 8=64$."},{"text":"32","correct":false,"explanation":"This halves 64 instead of finding its square root. A square root must multiply by itself to give 64, and $8 \\times 8=64$."}]},{"id":"be75367e-ca61-3806-34c9-853d56cc9959","cardType":"cloze","points":null,"front":"Squaring and square rooting are [ ] operations that undo each other.","back":"Squaring and square rooting are opposite operations that undo each other.","mcqOptions":[{"text":"opposite","correct":true,"explanation":"Squaring a positive number and then taking its square root returns the original number. The two operations therefore undo each other and are described here as opposite operations."},{"text":"equivalent","correct":false,"explanation":"Equivalent operations produce the same result, but squaring and square rooting generally do not. They are opposite operations because each reverses the effect of the other."},{"text":"identical","correct":false,"explanation":"Identical operations would perform the same mathematical process. Squaring multiplies a number by itself, while square rooting reverses that process, so they are opposite operations."},{"text":"commutative","correct":false,"explanation":"Commutative describes operations whose inputs can change order without changing the result, such as addition. Squaring and square rooting instead undo each other, so they are opposite operations."}]},{"id":"3cf37443-88f1-b6ca-279d-73a9d168a19b","cardType":"basic","points":1,"front":"What is a cube number?","back":"The answer when a number is multiplied by itself three times.","mcqOptions":[{"text":"The answer when a number is multiplied by itself three times.","correct":true,"explanation":"Cubing uses three equal factors: $n^3=n \\times n \\times n$. The result of multiplying a number by itself three times in this sense is therefore a cube number."},{"text":"The answer when a number is multiplied by itself two times.","correct":false,"explanation":"Using two equal factors gives a square number, such as $5^2=5 \\times 5=25$. A cube number uses three equal factors: $n \\times n \\times n$."},{"text":"The answer when a number is added to itself three times.","correct":false,"explanation":"Repeated addition produces a multiple, not a cube number. A cube is formed using multiplication with three equal factors, written $n^3=n \\times n \\times n$."},{"text":"The answer when a number is multiplied by three once.","correct":false,"explanation":"Multiplying by 3 produces a multiple of 3 and does not usually produce a cube. A cube number is formed from three equal factors: $n \\times n \\times n$."}]},{"id":"b7e3ac36-1f67-30fc-2bc2-cf6adcf4e4e7","cardType":"basic","points":1,"front":"Evaluate $4^3$.","back":"64","mcqOptions":[{"text":"64","correct":true,"explanation":"The exponent 3 means that three factors of 4 are multiplied together. Therefore, $4^3=4 \\times 4 \\times 4=16 \\times 4=64$."},{"text":"12","correct":false,"explanation":"This comes from multiplying the base by the exponent, $4 \\times 3$, rather than using repeated multiplication. Cubing gives $4 \\times 4 \\times 4=64$."},{"text":"16","correct":false,"explanation":"This is $4^2$, so it uses only two factors of 4. The exponent 3 requires three factors, giving $4 \\times 4 \\times 4=64$."},{"text":"81","correct":false,"explanation":"This is $3^4$, produced by swapping the base and exponent. The expression $4^3$ uses three factors of 4, so its value is 64."}]},{"id":"3d0057a7-9997-4485-fbb9-9d9c860d435c","cardType":"cloze","points":null,"front":"The mathematical symbol for the cube root of 125 is [ ].","back":"The mathematical symbol for the cube root of 125 is $\\sqrt[3]{125}$.","mcqOptions":[{"text":"$$\\sqrt[3]{125}$","correct":true,"explanation":"The small 3 is the root index and indicates a cube root. Therefore, the cube root of 125 is written $\\sqrt[3]{125}$."},{"text":"$$\\sqrt{125}$","correct":false,"explanation":"A radical sign without an index represents a square root, not a cube root. The required index is 3, so the correct symbol is $\\sqrt[3]{125}$."},{"text":"$$\\sqrt[125]{3}$","correct":false,"explanation":"This swaps the root index and the number being rooted, so it means the 125th root of 3. The cube root of 125 is $\\sqrt[3]{125}$."},{"text":"$$3\\sqrt{125}$","correct":false,"explanation":"Placing 3 before the radical means three times the square root of 125. For a cube root, 3 must be the index: $\\sqrt[3]{125}$."}]},{"id":"66e0ff09-9fa2-8dc4-05f7-9c53fe760a69","cardType":"basic","points":1,"front":"Evaluate $\\sqrt[3]{27}$.","back":"3","mcqOptions":[{"text":"3","correct":true,"explanation":"A cube root is the number that gives the original value when cubed. Since $3 \\times 3 \\times 3=27$, $\\sqrt[3]{27}=3$."},{"text":"9","correct":false,"explanation":"This comes from dividing 27 by 3, but cube rooting is not division by the root index. The correct answer is 3 because $3^3=27$."},{"text":"6","correct":false,"explanation":"This may come from confusing the cube root with an unrelated division or subtraction. Since $6^3=216$, the correct answer is 3 because $3^3=27$."},{"text":"-3","correct":false,"explanation":"Cubing $-3$ gives $-27$, not 27, because three negative factors produce a negative result. The cube root of positive 27 is therefore 3."}]},{"id":"6f75fbd4-d149-b992-b568-9ceece66e2a7","cardType":"basic","points":2,"front":"Why is the cube root of 27 not equal to 9?","back":"Cube root is not dividing by 3; it is the inverse of cubing ($3 \\times 3 \\times 3 = 27$).","mcqOptions":[{"text":"Cube root is not dividing by 3; it is the inverse of cubing ($3 \\times 3 \\times 3 = 27$).","correct":true,"explanation":"A cube root reverses cubing, so it asks which number appears as three equal factors. Because $3^3=27$, the cube root is 3 rather than $27 \\div 3=9$."},{"text":"Cube root is not dividing by 3; it is the inverse of squaring ($\\sqrt{27} \\times \\sqrt{27} = 27$).","correct":false,"explanation":"The inverse of squaring is square rooting, not cube rooting. Cube rooting reverses cubing, and $3 \\times 3 \\times 3=27$, so the cube root is 3."},{"text":"Cube root is not dividing by 3; it is the inverse of multiplying by 3 ($9 \\times 3 = 27$).","correct":false,"explanation":"The inverse of multiplying by 3 is dividing by 3, which gives 9 but does not find a cube root. Cube rooting reverses cubing, so $\\sqrt[3]{27}=3$."},{"text":"Cube root is not dividing by 3; it means finding three equal addends ($9 + 9 + 9 = 27$).","correct":false,"explanation":"Finding three equal addends uses repeated addition and gives one third of 27. A cube root requires three equal factors, and $3 \\times 3 \\times 3=27$."}]},{"id":"278e452f-d60c-6122-9a23-908264dbea20","cardType":"basic","points":2,"front":"How can you determine the value of $10^4$ by looking at the index?","back":"The index 4 tells you there are four zeros after the 1 (10,000).","mcqOptions":[{"text":"The index 4 tells you there are four zeros after the 1 (10,000).","correct":true,"explanation":"A positive integer power of 10 gives 1 followed by as many zeros as the index. Therefore, $10^4$ is 1 followed by four zeros."},{"text":"The index 4 tells you there are three zeros after the 1 (1,000).","correct":false,"explanation":"This counts the initial factor of 10 as one of the four multiplications. Since $10^4=10\\times10\\times10\\times10$, the result has four zeros: 10,000."},{"text":"The index 4 tells you to multiply 10 by 4 (40).","correct":false,"explanation":"An index indicates repeated multiplication, not multiplication by the index. Multiplying four factors of 10 gives $10^4=10,000$, which has four zeros after the 1."},{"text":"The index 4 tells you there are four zeros before the 1 (0.0001).","correct":false,"explanation":"This confuses a positive index with a negative index: $0.0001$ is $10^{-4}$. A positive index of 4 gives 10,000, with four zeros after the 1."}]},{"id":"e4d2de12-3c43-a654-4ac9-69529d183a9f","cardType":"basic","points":1,"front":"Evaluate $6 \\times 10^3$.","back":"6000","mcqOptions":[{"text":"6000","correct":true,"explanation":"Since $10^3=1000$, multiplying by 6 gives $6\\times1000=6000$. Equivalently, multiplying by $10^3$ moves the digits three places to the left."},{"text":"600","correct":false,"explanation":"This uses $10^2=100$ rather than $10^3=1000$. Multiplying 6 by the correct power gives $6\\times1000=6000$."},{"text":"60000","correct":false,"explanation":"This adds four zeros, treating the power as $10^4$. Since the index is 3, $10^3=1000$, so $6\\times10^3=6000$."},{"text":"18","correct":false,"explanation":"This multiplies 6 by the index 3 instead of evaluating the power. The index means $10^3=1000$, so the correct product is 6000."}]},{"id":"4b508eb2-89fa-29c3-0071-f317d9a1d2ff","cardType":"basic","points":1,"front":"List the powers of 2 from $2^1$ to $2^6$.","back":"2, 4, 8, 16, 32, 64","mcqOptions":[{"text":"2, 4, 8, 16, 32, 64","correct":true,"explanation":"Each new power of 2 is found by multiplying the previous value by 2. Starting with $2^1=2$ gives the sequence ending with $2^6=64$."},{"text":"2, 4, 6, 8, 10, 12","correct":false,"explanation":"These are the first six positive multiples of 2, not powers of 2. Powers are formed by repeated multiplication, giving 2, 4, 8, 16, 32, 64."},{"text":"1, 2, 4, 8, 16, 32","correct":false,"explanation":"This sequence begins with $2^0=1$, creating an off-by-one error. Starting at $2^1$ gives 2, 4, 8, 16, 32, 64."},{"text":"2, 4, 8, 16, 32, 128","correct":false,"explanation":"This skips $2^6=64$ and finishes with $2^7=128$. Doubling each term in order gives 2, 4, 8, 16, 32, 64."}]},{"id":"89a79b2b-fd45-1de0-f68e-aba283d5e6f6","cardType":"basic","points":1,"front":"List the first five cube numbers.","back":"1, 8, 27, 64, 125","mcqOptions":[{"text":"1, 8, 27, 64, 125","correct":true,"explanation":"Cube numbers are formed by multiplying each positive integer by itself three times. Thus $1^3$, $2^3$, $3^3$, $4^3$ and $5^3$ give these values."},{"text":"1, 4, 9, 16, 25","correct":false,"explanation":"These are square numbers, found using $n^2$, rather than cube numbers found using $n^3$. The first five positive cubes are 1, 8, 27, 64, 125."},{"text":"1, 8, 27, 64, 216","correct":false,"explanation":"The final value is $6^3=216$, so this skips $5^3=125$. The first five positive cubes end with 125, not 216."},{"text":"1, 8, 27, 81, 125","correct":false,"explanation":"This incorrectly uses 81 for the fourth cube, possibly confusing $4^3$ with $3^4$. Since $4\\times4\\times4=64$, the correct fourth term is 64."}]},{"id":"13a1e0b0-9212-f611-9152-9582d8591818","cardType":"basic","points":1,"front":"From the list [4, 8, 9, 16, 27, 64], which number is both a square number and a cube number?","back":"64 ($8^2$ and $4^3$)","mcqOptions":[{"text":"64 ($8^2$ and $4^3$)","correct":true,"explanation":"The number 64 is both a perfect square and a perfect cube because $8^2=64$ and $4^3=64$. It is the only such number in the list."},{"text":"16 ($4^2$ and $2^4$)","correct":false,"explanation":"Although 16 is a square, writing it as $2^4$ makes it a fourth power, not a cube. The number 64 satisfies both conditions as $8^2$ and $4^3$."},{"text":"8 ($4^2$ and $2^3$)","correct":false,"explanation":"The number 8 is a cube because $2^3=8$, but it is not the square of an integer; $4^2$ is 16. The required number is 64."},{"text":"27 ($3^2$ and $3^3$)","correct":false,"explanation":"The number 27 is a cube because $3^3=27$, but $3^2=9$, so it is not also a square. The number 64 is both $8^2$ and $4^3$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Powers and Roots","title":"Powers and Roots Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"35880eb9-7817-4b7c-a797-e24b2be85afd","walled":"$L39"}],"$L3a","$L3b"]