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SAT Mathematics : Solving Problems with Roots

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Example Questions

Example Question #21 : Exponents & Roots

$\frac{5x^{2}}{\sqrt[3]{x^{2}}}=$

Possible Answers:

$125x^{4}$

$5x\sqrt[3]{x}$

$5\sqrt[3]{x}$

$5x\sqrt[3]{x^{2}}$

Correct answer:

$5x\sqrt[3]{x}$

Explanation:

This problem emphasizes an important concept about mixing exponents and roots: when you see exponents and roots mixed, be ready to treat roots as fractional exponents. That way all of your terms are in a common form, and you can directly apply exponent rules to all terms.

Here, that means that you'll treat $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ . With that, the question then becomes:

$\frac{5x^{2}}{x^{\frac{2}{3}}}=$

Now you can use the exponent rule that $\frac{a^{b}}{a^{c}}=a^{b-c}$ (when you divide exponents of the same base, you subtract the exponents) to turn this into:

$5x^{(2-\frac{2}{3})}=5^{\frac{4}{3}}$

While you might now look to immediately convert back to the form of the answer choices (which each use radical signs to represent roots), $\sqrt[3]{x^{4}}$ is not an option. Which brings up another important point about mixing roots with exponents: generally when you're in exponent form you should stay there as long as possible, as the exponential form tends to provide you with more algebraic flexibility.

Here with $5x^{\frac{4}{3}}$ , you can pull out the $x^{1}$ (or just ) to break apart the mixed-number exponent. You can rephrase this as $5x(x^{\frac{1}{3}})$ , which then allows you to convert to an expression that looks like the answer choices. With the only fractional exponent left, $x^{\frac{1}{3}}$ , translating to $\sqrt[3]{x}$ , you can re-express the entire expression as $5x\sqrt[3]{x}$ .

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Example Question #22 : Exponents & Roots

$\sqrt[3]{\frac{2}{3a}}=$

Possible Answers:

$\frac{\sqrt[3]{18a^{2}}}{3a}$

$\frac{\sqrt[3]{6a}}{3a}$

Correct answer:

$\frac{\sqrt[3]{18a^{2}}}{3a}$

Explanation:

This problem creates confusion for people because they are not comfortable rationalizing the cube root in the denominator. To see how to do this properly, first view the given expression like this:

$\frac{\sqrt[3]{2}}{\sqrt[3]{3a}}$

and then ask what you would need to multiply the denominator by (and thus the numerator so that you're effectively multiplying by ) to make the expression under the root sign a perfect cube in the denominator. In this case, you must multiply top and bottom by $\sqrt[3]{9a^{2}}$ as shown below:

$\frac{\sqrt[3]{2}}{\sqrt[3]{3a}}\times\frac{\sqrt[3]{9a^{2}}}{\sqrt[3]{27a^{3}}}=\frac{\sqrt[3]{18a^{2}}}{\sqrt[3]{27a^{3}}}=\frac{\sqrt[3]{18a^{2}}}{3a}$ .

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Example Question #23 : Exponents & Roots

$\frac{\sqrt{6}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}}=$

Possible Answers:

$6\sqrt{3}-6\sqrt{2}$

$\frac{5}{6}$

$2\sqrt{3}+3\sqrt{2}$

$\frac{6}{5}$

Correct answer:

$6\sqrt{3}-6\sqrt{2}$

Explanation:

To simplify this multi-denominator expression, you should first multiply by one in the form $\frac{\sqrt{6}}{\sqrt{6}}$ to eliminate the two denominators in the bottom of the expression. This multiplication results in the following:

$\frac{\sqrt{6}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}}(\frac{\sqrt{6}}{\sqrt{6}})=\frac{6}{\frac{\sqrt{6}}{\sqrt{2}}+\frac{\sqrt{6}}{\sqrt{3}}}=\frac{6}{\sqrt{3}+\sqrt{2}}$

Now you must multiply by one again to remove the two roots in the denominator. To do that you must recognize the difference of squares and multiply the top and the bottom by the conjugate of $\sqrt{3}+\sqrt{2}$ as shown below:

$\frac{6}{\sqrt{3}+\sqrt{2}}(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}})=\frac{6\sqrt{3}-6\sqrt{2}}{3-2}=6\sqrt{3}-6\sqrt{2}$

Therefore, the correct answer is $6\sqrt{3}-6\sqrt{2}$ .

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Example Question #24 : Exponents & Roots

$(\sqrt{11}+\sqrt{11}+\sqrt{11})^{2}=$

Possible Answers:

121

Correct answer:

Explanation:

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or $y=\sqrt{(11)}$ s, or whatever). How many $\sqrt{(11)}$ ’s are there here? Three. This expression can be rewritten as $(3*\sqrt{(11)})^{2}$ . Now the exponent rules will apply; $(3*\sqrt{(11)})^{2}=3^{2}*(\sqrt{(11)})^{2}=9*11=99$ . The answer is .

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Example Question #25 : Exponents & Roots

Simplify:

$\frac{x-1}{4(\sqrt{x}+1)}$

Possible Answers:

$\frac{\sqrt{x}+1}{4}$

$\frac{\sqrt{x}-1}{4}$

$\frac{(x-1)\sqrt{x}-1}{4}$

$\frac{(x-1)\sqrt{x}+1}{4}$

Correct answer:

$\frac{\sqrt{x}-1}{4}$

Explanation:

To solve this problem we must recognize that (x-1) can be broken down into $(\sqrt{x}-1)(\sqrt{x}+1)$

After breaking (x-1) into $(\sqrt{x}-1)(\sqrt{x}+1)$ we see that the $(\sqrt{x}+1)$ terms in numerator and denominator cancel out.

This leaves us with $\frac{\sqrt{x}-1}{4}$

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SAT Mathematics : Solving Problems with Roots

Study concepts, example questions & explanations for SAT Mathematics

All SAT Mathematics Resources

Example Questions

Example Question #21 : Exponents & Roots

Example Question #22 : Exponents & Roots

Example Question #23 : Exponents & Roots

Example Question #24 : Exponents & Roots

Example Question #25 : Exponents & Roots

All SAT Mathematics Resources

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