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SAT Math : How to find patterns in exponents

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Example Question #571 : Algebra

Write in radical notation:

$\left ( x+3 \right )^{\frac{2}{3}}$

Possible Answers:

$\sqrt[3]{\left ( x+3 \right )}$

$\sqrt[]{\left ( x+3 \right )^{2}}$

$\sqrt[3]{\left ( x+3 \right )^{2}}$

$\sqrt[]{\left ( x+3 \right )}$

$x^{\frac{2}{3}} + 3^{\frac{2}{3}}$

Correct answer:

$\sqrt[3]{\left ( x+3 \right )^{2}}$

Explanation:

Properties of Radicals

$\sqrt[]{x} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{\frac{1}{3}}$

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Example Question #11 : How To Find Patterns In Exponents

Express in radical form : $x^{\frac{3}{5}}$

Possible Answers:

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

$x^{15}$

$x^{0.6}$

$\sqrt[3]{x}$

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

Correct answer:

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

Explanation:

Properties of Radicals

$\sqrt[]{x} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{\frac{1}{3}}$

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

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Example Question #572 : Algebra

Simplify: $\sqrt[3]{2}\star \sqrt[3]{32}$

Possible Answers:

$1.26\star 3.17$

$\sqrt[3]{\frac{32}{2}} = \sqrt[3]{16} = 2.52$

$\sqrt[3]{2}\star \sqrt[3]{32} = \sqrt[3]{64} = 4$

$8\star 96 = 768$

$\sqrt[3]{\frac{2}{32}} = \sqrt[3]{\frac{1}{16}} = \frac{1}{2.52}$

Correct answer:

$\sqrt[3]{2}\star \sqrt[3]{32} = \sqrt[3]{64} = 4$

Explanation:

$\sqrt[3]{2}\star \sqrt[3]{32}= \sqrt[3]{64}= 4$

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Example Question #265 : Exponents

Simplify:

$\frac{\sqrt[3]{x^{5}y^{-2}}}{\sqrt[3]{x^{-1}y^{4}}}$

Possible Answers:

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

$\frac{\sqrt[3]{x^{5}}}{\sqrt[3]{y^{4}}}$

$\frac{\sqrt[3]{y^{-2}}}{\sqrt[3]{x^{-1}}}$

$\frac{x^{15}y^{-6}}{x^{-3}y^{12}}$

$\frac{x^{2}}{y^{2}}$

Correct answer:

$\frac{x^{2}}{y^{2}}$

Explanation:

Convert the given expression into a single radical e.g. $\sqrt[3]{\frac{x^{5}y^{-2}}{x^{-1}y^{4}}}$ the expression inside the radical is:

$\frac{x^{6}}{y^{6}}$

and the cube root of this is :

$\frac{x^{2}}{y^{2}}$

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Example Question #261 : Exponents

Solve for .

$\log _{\frac{1}{5}}\left ( \frac{1}{25} \right )=x$

Possible Answers:

Correct answer:

Explanation:

$\frac{1}{25}= \left ( \frac{1}{5} \right )^{x}$

Hence must be equal to 2.

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Example Question #267 : Exponents

Simplify:

$log\left (\frac{x}{y^{2}} \right )$

Possible Answers:

$logx+logy^{2}$

logx-2logy

2logy-logx

$\left ( logx \right )\left ( logy^{2} \right )$

$logx-logy^{2}$

Correct answer:

logx-2logy

Explanation:

$log\left ( \frac{x}{y^{2}} \right )=logx-logy^{2}$

Now $logy^{2} = 2logy$

Hence the correct answer is logx - 2logy

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Example Question #268 : Exponents

Solve for .

$\ln x=\ln (x-1)-\ln (x+3)$

Possible Answers:

$no\ solution$

Correct answer:

Explanation:

If we combine into a single logarithmic function we get:

$\ln x=\ln \frac{x-1}{x+3}$

$x = \frac{x-1}{x+3}$

Solving for we get x=-1 .

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Example Question #269 : Exponents

If is the complex number such that i^2=-1 , evaluate the following expression:

$i^{25}-i^{23}+i^{21}-i^{19}+i^{17}-i^{15}.\ .\ .+i$

Possible Answers:

13i

-25i

12i

Correct answer:

13i

Explanation:

The powers of i form a sequence that repeats every four terms.

i¹= i

i² = -1

i³ = -i

i⁴ = 1

i⁵ = i

Thus:

i²⁵ = i

i²³ = -i

i²¹ = i

i¹⁹= -i

Now we can evalulate the expression.

i²⁵ - i²³+ i²¹- i¹⁹ + i¹⁷..... + i

= i + (-1)(-i) + i + (-1)(i) ..... + i

= i + i + i + i + ..... + i

Each term reduces to +i. Since there are 13 terms in the expression, the final result is 13i.

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Example Question #12 : Exponents

If $x^7y^8z^{10} < 0$ , then which of the following must also be true?

Possible Answers:

y < 0

zx > 0

x < 0

yz < 0

zy > 0

Correct answer:

x < 0

Explanation:

We know that the expression must be negative. Therefore one or all of the terms x⁷, y⁸ and z¹⁰ must be negative; however, even powers always produce positive numbers, so y⁸ and z¹⁰ will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x⁷ must be negative, so x must be negative. Thus, the answer is x < 0.

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Example Question #1 : How To Find Patterns In Exponents

Simplify the following:

$\frac{48^5^0+80^3^0}{4^2^0}$

Possible Answers:

17^5^0

2^1^6^0*3^5^0+2^8^0*5^3^0

4^1^6*3^2^5+2^2^0*5^1^5

12^3^0+5^1^0

17^8^0

Correct answer:

2^1^6^0*3^5^0+2^8^0*5^3^0

Explanation:

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

48^5^0+80^3^0=(2^4*3)^5^0+(2^4*5)^3^0

Continuing the simplification:

(2^4*3)^5^0+(2^4*5)^3^0=2^2^0^0*3^5^0+2^1^2^0*5^3^0

Now, these factors have in common a 2^1^2^0 . Factor this out:

2^1^2^0(2^8^0*3^5^0+5^3^0)

Denominator

This is much simpler:

4^2^0=(2^2)^2^0=2^4^0

Now, return to your fraction:

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}$

Cancel out the common factors of :

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}=2^8^0(2^8^0*3^5^0+5^3^0)=2^1^6^0*3^5^0+2^8^0*5^3^0$

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SAT Math : How to find patterns in exponents

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #571 : Algebra

Example Question #11 : How To Find Patterns In Exponents

Example Question #572 : Algebra

Example Question #265 : Exponents

Example Question #261 : Exponents

Example Question #267 : Exponents

Example Question #268 : Exponents

Example Question #269 : Exponents

Example Question #12 : Exponents

Example Question #1 : How To Find Patterns In Exponents

All SAT Math Resources

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