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High School Math : Understanding Exponents

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Example Questions

Example Question #1 : Understanding Negative Exponents

Which of the following is equivalent to $3^{-2}$ $\displaystyle 3^{-2}$ ?

Possible Answers:

$\frac{1}{6}$ $\displaystyle \frac{1}{6}$

$\displaystyle -6$

$\frac{1}{9}$ $\displaystyle \frac{1}{9}$

$\displaystyle 1$

$\displaystyle -9$

Correct answer:

$\frac{1}{9}$ $\displaystyle \frac{1}{9}$

Explanation:

By definition,

$b^{-x} = \frac{1}{b^{x}}$ $\displaystyle b^{-x} = \frac{1}{b^{x}}$ .

In our problem, b = 3 $\displaystyle b = 3$ and x = 2 $\displaystyle x = 2$ .

Then, we have $\frac{1}{3^{2}} = \frac{1}{9}$ $\displaystyle \frac{1}{3^{2}} = \frac{1}{9}$ .

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Example Question #1 : Exponents

Solve for $\displaystyle x$ :

$(x+5)^{-3} = -1$ $\displaystyle (x+5)^{-3} = -1$

Possible Answers:

$\displaystyle -5$

$\displaystyle -1$

$\displaystyle -6$

$\displaystyle -3$

$\displaystyle -4$

Correct answer:

$\displaystyle -6$

Explanation:

Raise both sides of the equation to the inverse power of $\displaystyle -3$ to cancel the exponent on the left hand side of the equation.

$\rightarrow ((x+5)^{-3})^{-\frac{1}{3}} = (-1)^{-\frac{1}{3}}$ $\displaystyle \rightarrow ((x+5)^{-3})^{-\frac{1}{3}} = (-1)^{-\frac{1}{3}}$

$\rightarrow x+5 = -1$ $\displaystyle \rightarrow x+5 = -1$

Subtract $\displaystyle 5$ from both sides:

$\rightarrow (x+5) - 5 = (-1)-5$ $\displaystyle \rightarrow (x+5) - 5 = (-1)-5$

$\rightarrow x = -6$ $\displaystyle \rightarrow x = -6$

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Example Question #1 : Understanding Fractional Exponents

Convert the exponent to radical notation.

$x^{\frac{3}{7}}$ $\displaystyle x^{\frac{3}{7}}$

Possible Answers:

$\small \small \sqrt[7]{x^3}$ $\displaystyle \small \small \sqrt[7]{x^3}$

$\small \small \sqrt[3]{x^7}$ $\displaystyle \small \small \sqrt[3]{x^7}$

$\small \frac{1}{x^4}$ $\displaystyle \small \frac{1}{x^4}$

$\small \frac{x^3}{x^7}$ $\displaystyle \small \frac{x^3}{x^7}$

Correct answer:

$\small \small \sqrt[7]{x^3}$ $\displaystyle \small \small \sqrt[7]{x^3}$

Explanation:

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$ $\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}$

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$ $\displaystyle x^{\frac{3}{7}}=\sqrt[7]{x^3}$

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Example Question #1 : Exponents

Which of the following is equivalent to $64^{\frac{1}{2}}$ $\displaystyle 64^{\frac{1}{2}}$ ?

Possible Answers:

$\frac{1}{32}$ $\displaystyle \frac{1}{32}$

$\displaystyle 32$

$\displaystyle 8$

$\displaystyle 16$

$\displaystyle -32$

Correct answer:

$\displaystyle 8$

Explanation:

By definition, a number raised to the $\frac{1}{2}$ $\displaystyle \frac{1}{2}$ power is the same as the square root of that number.

Since the square root of 64 is 8, 8 is our solution.

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Example Question #1 : Fractional Exponents

Simplify the expression:

$\small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$ $\displaystyle \small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$

Possible Answers:

256 $\displaystyle 256$

$\displaystyle 16$

$\displaystyle 64$

1024 $\displaystyle 1024$

Correct answer:

256 $\displaystyle 256$

Explanation:

Remember that fraction exponents are the same as radicals.

$\small 16^{\frac{1}{2}}=\sqrt{16}=4$ $\displaystyle \small 16^{\frac{1}{2}}=\sqrt{16}=4$

$256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$ $\displaystyle 256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

$16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4$ $\displaystyle 16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4$

$\small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$ $\displaystyle \small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$

Either method, we then need to multiply to two terms.

$\small (4)(64)=256$ $\displaystyle \small (4)(64)=256$

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High School Math : Understanding Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Negative Exponents

Example Question #1 : Exponents

Example Question #1 : Understanding Fractional Exponents

Example Question #1 : Exponents

Example Question #1 : Fractional Exponents

All High School Math Resources

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