Solving Other Functions - SAT Math

Card 0 of 8

Question

Simplify:

$\sqrt[3]{\sqrt[4]{x^{6}}}$

You may assume that is a nonnegative real number.

Answer

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

$\sqrt[3]{\sqrt[4]{x^{6}}} = \sqrt[3]{ \left (x^{6} \right ) ^{\frac{1}{4}}} =\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}}}$

Multiply the exponents, per the power of a power rule:

$\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}} = x ^{6 \cdot \frac{1}{4} \cdot \frac{1}{3}} = x^{\frac{1}{2}} = \sqrt{x}$

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Question

Define functions $p(x) = x^{5} + 2x^{2} - 7$ and $q(x)= x ^{4} - 3x$ .

for exactly one value of on the interval .

Which of the following statements is correct about ?

Answer

Define

$= (x^{5} + 2x^{2} - 7) - (x ^{4} - 3x )$

$= x^{5} - x ^{4}+ 2x^{2}+ 3x - 7$

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some $c \in (a, b)$ . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values: $\left { 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\right }$ :

$g(1.0)= 1.0^{5} - 1.0 ^{4}+ 2 \cdot 1.0^{2}+ 3 \cdot 1.0 - 7 \approx -2$

$g(1.1)= 1.1^{5} - 1.1 ^{4}+ 2 \cdot 1.1^{2}+ 3 \cdot 1.1 - 7 \approx - 1.13$

$g(1.2)= 1.2^{5} - 1.2 ^{4}+ 2 \cdot 1.2^{2}+ 3 \cdot 1.2 - 7 \approx -0.11$

$g(1.3)= 1.3^{5} - 1.3 ^{4}+ 2 \cdot 1.3^{2}+ 3 \cdot 1.3 - 7 \approx 1.4$

$g(1.4)= 1.4^{5} - 1.4 ^{4}+ 2 \cdot 1.4^{2}+ 3 \cdot 1.4 - 7 \approx 2.66$

$g(1.5)= 1.5^{5} - 1.5 ^{4}+ 2 \cdot 1.5^{2}+ 3 \cdot 1.5 - 7 \approx 4.53$

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some $c \in (1.1, 1.2)$ .

Compare your answer with the correct one above

Question

Simplify:

$\sqrt[3]{\sqrt[4]{x^{6}}}$

You may assume that is a nonnegative real number.

Answer

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

$\sqrt[3]{\sqrt[4]{x^{6}}} = \sqrt[3]{ \left (x^{6} \right ) ^{\frac{1}{4}}} =\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}}}$

Multiply the exponents, per the power of a power rule:

$\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}} = x ^{6 \cdot \frac{1}{4} \cdot \frac{1}{3}} = x^{\frac{1}{2}} = \sqrt{x}$

Compare your answer with the correct one above

Question

Define functions $p(x) = x^{5} + 2x^{2} - 7$ and $q(x)= x ^{4} - 3x$ .

for exactly one value of on the interval .

Which of the following statements is correct about ?

Answer

Define

$= (x^{5} + 2x^{2} - 7) - (x ^{4} - 3x )$

$= x^{5} - x ^{4}+ 2x^{2}+ 3x - 7$

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some $c \in (a, b)$ . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values: $\left { 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\right }$ :

$g(1.0)= 1.0^{5} - 1.0 ^{4}+ 2 \cdot 1.0^{2}+ 3 \cdot 1.0 - 7 \approx -2$

$g(1.1)= 1.1^{5} - 1.1 ^{4}+ 2 \cdot 1.1^{2}+ 3 \cdot 1.1 - 7 \approx - 1.13$

$g(1.2)= 1.2^{5} - 1.2 ^{4}+ 2 \cdot 1.2^{2}+ 3 \cdot 1.2 - 7 \approx -0.11$

$g(1.3)= 1.3^{5} - 1.3 ^{4}+ 2 \cdot 1.3^{2}+ 3 \cdot 1.3 - 7 \approx 1.4$

$g(1.4)= 1.4^{5} - 1.4 ^{4}+ 2 \cdot 1.4^{2}+ 3 \cdot 1.4 - 7 \approx 2.66$

$g(1.5)= 1.5^{5} - 1.5 ^{4}+ 2 \cdot 1.5^{2}+ 3 \cdot 1.5 - 7 \approx 4.53$

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some $c \in (1.1, 1.2)$ .

Compare your answer with the correct one above

Question

Simplify:

$\sqrt[3]{\sqrt[4]{x^{6}}}$

You may assume that is a nonnegative real number.

Answer

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

$\sqrt[3]{\sqrt[4]{x^{6}}} = \sqrt[3]{ \left (x^{6} \right ) ^{\frac{1}{4}}} =\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}}}$

Multiply the exponents, per the power of a power rule:

$\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}} = x ^{6 \cdot \frac{1}{4} \cdot \frac{1}{3}} = x^{\frac{1}{2}} = \sqrt{x}$

Compare your answer with the correct one above

Question

Define functions $p(x) = x^{5} + 2x^{2} - 7$ and $q(x)= x ^{4} - 3x$ .

for exactly one value of on the interval .

Which of the following statements is correct about ?

Answer

Define

$= (x^{5} + 2x^{2} - 7) - (x ^{4} - 3x )$

$= x^{5} - x ^{4}+ 2x^{2}+ 3x - 7$

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some $c \in (a, b)$ . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values: $\left { 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\right }$ :

$g(1.0)= 1.0^{5} - 1.0 ^{4}+ 2 \cdot 1.0^{2}+ 3 \cdot 1.0 - 7 \approx -2$

$g(1.1)= 1.1^{5} - 1.1 ^{4}+ 2 \cdot 1.1^{2}+ 3 \cdot 1.1 - 7 \approx - 1.13$

$g(1.2)= 1.2^{5} - 1.2 ^{4}+ 2 \cdot 1.2^{2}+ 3 \cdot 1.2 - 7 \approx -0.11$

$g(1.3)= 1.3^{5} - 1.3 ^{4}+ 2 \cdot 1.3^{2}+ 3 \cdot 1.3 - 7 \approx 1.4$

$g(1.4)= 1.4^{5} - 1.4 ^{4}+ 2 \cdot 1.4^{2}+ 3 \cdot 1.4 - 7 \approx 2.66$

$g(1.5)= 1.5^{5} - 1.5 ^{4}+ 2 \cdot 1.5^{2}+ 3 \cdot 1.5 - 7 \approx 4.53$

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some $c \in (1.1, 1.2)$ .

Compare your answer with the correct one above

Question

Simplify:

$\sqrt[3]{\sqrt[4]{x^{6}}}$

You may assume that is a nonnegative real number.

Answer

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

$\sqrt[3]{\sqrt[4]{x^{6}}} = \sqrt[3]{ \left (x^{6} \right ) ^{\frac{1}{4}}} =\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}}}$

Multiply the exponents, per the power of a power rule:

$\left [ \left (x^{6} \right ) ^{\frac{1}{4} \right ] ^ {\frac{1}{3}} = x ^{6 \cdot \frac{1}{4} \cdot \frac{1}{3}} = x^{\frac{1}{2}} = \sqrt{x}$

Compare your answer with the correct one above

Question

Define functions $p(x) = x^{5} + 2x^{2} - 7$ and $q(x)= x ^{4} - 3x$ .

for exactly one value of on the interval .

Which of the following statements is correct about ?

Answer

Define

$= (x^{5} + 2x^{2} - 7) - (x ^{4} - 3x )$

$= x^{5} - x ^{4}+ 2x^{2}+ 3x - 7$

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some $c \in (a, b)$ . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values: $\left { 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\right }$ :

$g(1.0)= 1.0^{5} - 1.0 ^{4}+ 2 \cdot 1.0^{2}+ 3 \cdot 1.0 - 7 \approx -2$

$g(1.1)= 1.1^{5} - 1.1 ^{4}+ 2 \cdot 1.1^{2}+ 3 \cdot 1.1 - 7 \approx - 1.13$

$g(1.2)= 1.2^{5} - 1.2 ^{4}+ 2 \cdot 1.2^{2}+ 3 \cdot 1.2 - 7 \approx -0.11$

$g(1.3)= 1.3^{5} - 1.3 ^{4}+ 2 \cdot 1.3^{2}+ 3 \cdot 1.3 - 7 \approx 1.4$

$g(1.4)= 1.4^{5} - 1.4 ^{4}+ 2 \cdot 1.4^{2}+ 3 \cdot 1.4 - 7 \approx 2.66$

$g(1.5)= 1.5^{5} - 1.5 ^{4}+ 2 \cdot 1.5^{2}+ 3 \cdot 1.5 - 7 \approx 4.53$

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some $c \in (1.1, 1.2)$ .

Compare your answer with the correct one above

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