Understanding Exponents

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Algebra II › Understanding Exponents

Questions 1 - 10

Simplify .

Explanation

To solve this expression, remove the outer exponent and expand the terms.

$(x^2y^3)^3=x^2\cdot x^2 \cdot x^2 \cdot y^3 \cdot y^3 \cdot y^3$

By exponential rules, add all the powers when multiplying like terms.

The answer is:

Expand:

Explanation

When we expand exponents, we simply repeat the base by the exponential value.

Therefore:

Evaluate: $7^{-3}$

$\frac{1}{7^3}$

$-\frac{1}{7^3}$

Explanation

When dealing with negative exponents, we write $x^{-a}=\frac{1}{x^a}$ . Therefore $7^{-3}=\frac{1}{7^3}$ .

Expand

Explanation

When expanding exponents, we repeat the base by the exponential value.

Expand

Explanation

To expand the exponent, we multiply the base by the power it is being raised to.

Solve for :

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

Explanation

Raise both sides of the equation to the inverse power of to cancel the exponent on the left hand side of the equation.

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

$\rightarrow x+5 = -1$

Subtract from both sides:

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

$\rightarrow x = -6$

Evaluate: $3^{-4}$

$\frac{1}{81}$

$-\frac{1}{81}$

Explanation

When dealing with exponents, we convert as such: $x^{-a}=\frac{1}{x^a}$ . Therefore, $3^{-4}=\frac{1}{3^4}=\frac{1}{81}$ .

Evaluate: $5^{-8}$

$\frac{1}{5^8}$

$-\frac{1}{5^8}$

$\frac{1}{5^{-8}}$

$-5^{-8}$

Explanation

When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$5^{-8}=\frac{1}{5^8}$

Simplify:

$10^{-5}$

$\frac{1}{10^5}$

$-\frac{1}{10^5}$

Explanation

When an exponent is negative, we express as such:

$x^{-a}=\frac{1}{x^a}$

is the positive exponent, and is the base.

$10^{-5}=\frac{1}{10^5}= \frac{1}{100000}$ .

Evaluate $2^{\frac{1}2{}}$

$\sqrt{2}$

$\frac{1}{4}$

$\frac{\sqrt{2}}2{}$

Explanation

When dealing with fractional exponents, remember this form:

$\sqrt[a]{x^n}$ is the index of the radical which is also the denominator of the fraction, represents the base of the exponent, and is the power the base is raised to. That value is the numerator of the fraction.

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

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