# SAT Exponents and Radicals Worksheet PDF with Answers Source: https://1600.now/sat-equivalent-expressions-worksheet --- ## SAT Exponents and Radicals Worksheet Multiplying powers with the same base adds exponents; raising a power to another power multiplies them. A negative exponent means a reciprocal. These rules apply to products, not to separate terms across a plus sign. Written by [Luke Finigan](https://1600.now/about) The 12 exponents-and-radicals questions below also cover binomial squares, factoring, and rational expressions. The printable student copy hides the answers; the full PDF includes explanations. Choose Letter or A4. 12 questions · about 25 minutes · Advanced Math › Equivalent expressions [Download 4-page PDF](https://1600.now/downloads/worksheets/sat-equivalent-expressions-worksheet-student-letter.pdf) [Share to Google Classroom](https://classroom.google.com/share?url=https%3A%2F%2F1600.now%2Fsat-equivalent-expressions-worksheet) · This worksheet is free to print and share under [CC BY-NC-ND 4.0](https://creativecommons.org/licenses/by-nc-nd/4.0/). Tutors may also use it in paid sessions. If you post it online, please link to this page instead of re-uploading the PDF. These are original questions, not College Board items. ### Check equivalence without losing terms For nonzero $x$, $x^a x^b=x^{a+b}$, $\dfrac{x^a}{x^b}=x^{a-b}$, and $(x^a)^b=x^{ab}$. If the power surrounds a product, apply it to each factor: $(2x^3)^2=4x^6$. Simplify a radical by separating a perfect-square factor: $\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2$. Squaring the result gives $72$, which checks the simplification. Factor the entire numerator before canceling a rational expression. Keep every restriction from the original denominator. Testing one allowed value of $x$ can disprove an answer, but matching at one value does not prove equivalence; expand or factor to establish that. Try it yourself: For $x\ne2$, which expression equals $\dfrac{x^2-4}{x-2}$? · A $x-2$ B $x+2$ C $x$ ### Questions 1. **Question 1** For nonzero $x$, which expression equals $x^3$ multiplied by $x^5$? A. $x^2$ B. $x^8$ C. $x^{15}$ D. $2x^8$ **B.** Multiplication of powers with the same base adds the exponents: $3 + 5 = 8$, giving $x^8$. 2. **Question 2** For nonzero $a$, which expression equals $\dfrac{a^7}{a^2}$? A. $a^3$ B. $a^5$ C. $a^9$ D. $a^{14}$ **B.** For a quotient with the same nonzero base, subtract exponents: $7 - 2 = 5$. 3. **Question 3** Which expression equals $(2x^{3})^{2}$? A. $2x^{5}$ B. $2x^{6}$ C. $4x^{5}$ D. $4x^{6}$ **D.** Square both factors: $2^{2}=4$ and $(x^{3})^{2}=x^{6}$. Their product is $4x^{6}$. 4. **Question 4** What is $\sqrt{144}$? A. 12 B. 24 C. 72 D. 144 **A.** The square root denotes the nonnegative number whose square is 144. Since $12^{2}=144$, the value is 12. 5. **Question 5** Which expression equals $\sqrt{72}$? A. $6\sqrt{2}$ B. $8\sqrt{2}$ C. $3\sqrt{2}$ D. 36 **A.** Factor 72 as 36 times 2. Taking the square root gives $6\sqrt{2}$. 6. **Question 6** For positive $x$, which expression equals $\sqrt{x^{6}}$? A. $x^2$ B. $x^{3}$ C. $x^{6}$ D. $x^{12}$ **B.** The square root halves the exponent when $x$ is positive: $(x^{6})^{1/2}=x^{3}$. 7. **Question 7** If $2^{x}=32$, what is $x$? A. 4 B. 5 C. 8 D. 16 **B.** Repeated multiplication gives $2^{5}=32$. Therefore the exponent $x$ is 5. 8. **Question 8** For nonzero $x$, which expression equals $\dfrac{1}{x^{4}}$? A. $x^{-4}$ B. $-x^{4}$ C. $x^{-1}$ D. $\dfrac{4}{x}$ **A.** A negative exponent means reciprocal: $x^{-4}=\dfrac{1}{x^{4}}$. It does not mean that the value itself must be negative. 9. **Question 9** If $3^{2x}=81$, what is $x$? A. 1 B. 2 C. 3 D. 4 **B.** Since $81=3^{4}$, the exponents satisfy $2x=4$. Dividing by 2 gives $x=2$. 10. **Question 10** Which expression equals $(x+4)^{2}$? A. $x^{2}+16$ B. $x^{2}+4x+16$ C. $x^{2}+8x+16$ D. $2x+8$ **C.** Multiplying $(x+4)(x+4)$ produces $x^{2}+4x+4x+16$, which combines to $x^{2}+8x+16$. 11. **Question 11** Which expression equals $x^{2}-25$? A. $(x-5)^{2}$ B. $(x-5)(x+5)$ C. $(x-25)(x+1)$ D. $x(x-25)$ **B.** This is a difference of squares: $x^{2}-5^{2}=(x-5)(x+5)$. The middle terms cancel. 12. **Question 12** For $x$ not equal to 3, which expression equals $\dfrac{x^{2}-9}{x-3}$? A. $x-3$ B. $x+3$ C. $x^{2}+3$ D. 1 **B.** Factor the numerator as $(x-3)(x+3)$. Cancel $x-3$ only when $x$ is not 3, leaving $x+3$. 0/12 correct ### Check your exponent answers Each row is a wrong answer choice from one of the sheet's problems. Find the one you picked, then run the check in the last column. | What your answer looked like | What actually happened | Fix it next time | | --- | --- | --- | | $x^{15}$ for $x^3 \cdot x^5$ (problem 1) | You multiplied the exponents. That rule is for a power of a power, like $(x^3)^5 = x^{15}$. | Multiplying powers with the same base adds the exponents: 3 factors of $x$ times 5 more is 8 factors, so $x^8$. | | $2x^6$ for $(2x^3)^2$ (problem 3) | The square reached $x^3$ but skipped the 2. An exponent outside parentheses applies to every factor inside. | Write it as $(2x^3)(2x^3) = 4x^6$. Check with $x = 1$: $(2 \cdot 1^3)^2 = 4$, while $2x^6$ gives 2. | | 36 for $\sqrt{72}$ (problem 5) | You halved 72. A square root asks which number times itself gives 72, and $36 \cdot 36 = 1296$. | Pull out the largest perfect-square factor: $72 = 36 \cdot 2$, so $\sqrt{72} = 6\sqrt{2}$. Check: $(6\sqrt{2})^2 = 36 \cdot 2 = 72$. | | $-x^4$ for $\dfrac{1}{x^4}$ (problem 8) | A negative exponent means reciprocal, not a negative number: $x^{-4} = \dfrac{1}{x^4}$. | Test $x = 2$: $\dfrac{1}{2^4} = \dfrac{1}{16}$ is positive, but $-2^4 = -16$. The expression that matches is $x^{-4}$. | | $x = 4$ from $3^{2x} = 81$ (problem 9) | You solved for $2x$ and stopped. Writing $81 = 3^4$ gives $2x = 4$, and the question asks for $x$. | Divide by 2 to get $x = 2$, then plug it back in: $3^{2 \cdot 2} = 3^4 = 81$. With $x = 4$ you'd get $3^8 = 6561$. | | $x^2 + 16$ for $(x + 4)^2$ (problem 10) | You squared each term separately and lost the middle term. $(x + 4)^2$ means $(x + 4)(x + 4)$, which produces $4x$ twice. | Test $x = 1$: $(1 + 4)^2 = 25$, but $1^2 + 16 = 17$. The full expansion $x^2 + 8x + 16$ gives $1 + 8 + 16 = 25$. | | $(x - 5)^2$ for $x^2 - 25$ (problem 11) | $(x - 5)^2$ expands to $x^2 - 10x + 25$, which has a middle term and ends in $+25$. | A difference of squares factors into one minus and one plus: $(x - 5)(x + 5)$. Expand it to check: $-5x$ and $+5x$ cancel, leaving $x^2 - 25$. | #### Worked example: simplifying a fraction by factoring Problem 12 asks which expression equals $\dfrac{x^2 - 9}{x - 3}$ for $x \ne 3$. Dividing term by term, $\dfrac{x^2}{x} = x$ and $\dfrac{9}{3} = 3$, produces the choice $x - 3$, and it's wrong: you can cancel only a factor of the whole numerator against a factor of the whole denominator. So factor first. The numerator is a difference of squares, $x^2 - 3^2 = (x - 3)(x + 3)$, and cancelling the shared $x - 3$ leaves $x + 3$. The condition $x \ne 3$ is there because the original denominator is 0 at $x = 3$. Check with $x = 5$: $\dfrac{25 - 9}{5 - 3} = \dfrac{16}{2} = 8$, and $5 + 3 = 8$. The other choices give 2, 28, and 1. ### Apply the check to fresh questions Retry a missed expression by expanding both sides or factoring the original. Then test an allowed value as a check. The practice link gives more equivalent-expression questions. [Practice equivalent expressions questions](https://1600.now/bank/math/skill/Equivalent%20expressions) [Review the equivalent expressions method](https://1600.now/sat-skill/equivalent-expressions) ### More Math worksheets [All worksheets](https://1600.now/sat-worksheets) - [Quadratic Functions](https://1600.now/sat-nonlinear-functions-worksheet) - [Nonlinear Equations and Systems](https://1600.now/sat-nonlinear-equations-and-systems-worksheet) - [Linear Equations](https://1600.now/sat-linear-equations-one-variable-worksheet)