Real Exponents - College Algebra

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Question

Simplfy:

$\frac{x^{3.78}}{x^{2.58}}$

Answer

Treat this with regular exponent rules.

$\frac{x^{3.78}}{x^{2.58}}=x^{3.78-2.58}=x^{1.2}$

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Question

Simplify: $\frac{15x^2y^5c^3}{5xy^6c^3}$

Answer

$\frac{15x^2y^5c^3}{5xy^6c^3} \rightarrow 3xy^{-1}\rightarrow 3x*\frac{1}{y}=\boldsymbol{\frac{3x}{y}}$

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Question

Solve for :

$a^{2}\cdot a^{3}= 243$

Answer

When like bases with exponents are multiplied, the value of the product's exponent is the sum of both original exponents as shown here:

$a^{n}\cdot a^{m}=a^{n+m}$

We can use this common rule to solve for in the practice problem:

$\a^{2}\cdot a^{3}=243\a^{2}\cdot a^{3}=a^{2+3}\a^{5}=243$

$a=\sqrt[5]{243}\a=3$

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Question

Solve for :

$\frac{a^{7}}{a^3}=256$

Answer

The product of dividing like bases with exponents is the difference of the numerator and denominator exponents. This is a common rule when working with rational exponents:

$\frac{a^{m}}{a^{n}}=a^{m-n}$

We can use this common rule to solve for :

$\\frac{a^{7}}{a^{3}}=256\\a^{7-3}=256\a^{4}=256\a=\sqrt[4]{256}\a=4$

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Question

Solve for :

$3^{x-2}\cdot9^{2x}=27$

Answer

To solve for , we need all values to have like bases:

$\3^{x-2}\cdot9^{2x}=27\3^{x-2}\cdot3^{2\cdot2x}=27\3^{x-2}\cdot3^{4x}=3^{3}$

Now that all values have like bases, we can solve for :

$\x-2+4x=3\5x-2=3\5x=5\x=1$

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Question

Solve for :

$27^{\frac{x-3}{3}}\cdot3^{\frac{3x}{4}}=81$

Answer

To solve for , we want all values in the equation to have like bases:

$\27^{\frac{x-3}{3}}\cdot3^{\frac{3x}{4}}=81 \3^{3\cdot\frac{x-3}{3}}\cdot3^{\frac{3x}{4}}=3^{4} \3^{x-3}\cdot3^{\frac{3x}{4}}=3^{4}$

Now we can solve for :

$\x-3+\frac{3x}{4}=4\\frac{7x}{4}=7\7x=28\x=4$

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Question

Solve for :

$5^{\frac{x}{4}}\cdot5^{\frac{x}{8}}=125$

Answer

To solve for , we want all the values in the equation to have like bases:

$\5^{\frac{x}{4}}\cdot5^{\frac{x}{8}}=125 \5^{\frac{x}{4}}\cdot5^{\frac{x}{8}}=5^3$

Now we can solve for :

$\\frac{x}{4}+\frac{x}{8}=3 \8\cdot(\frac{x}{4}+\frac{x}{8})=3\cdot8 \2x+x=24\3x=24\x=8$

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Question

Simplify the following expression.

$(2x^{7.2})(x^{-6.2})$

Answer

When multiplying exponential, the exponents always add. While the 2 in the front of the first exponential might throw you off, you may disregard it initially.

$(2)(x^{7.2}x^{-6.2})$

Which simplifies to

$(2)x^{7.2-6.2}=(2)x^1$

Our final answer is

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Question

Simplify the following expression:

$\frac{(x^{2})^{5}x^{-8}}{x^2}$

Answer

First, we need to simplify the numerator. First term, can be simplified to $x^{10}$ . Plugging this back into the numerator, we get

$x^{10}x^{-8}$ , which simplifies to . Plugging this back into the original equation gives us

$\frac{x^2}{x^2}$ , which is simply .

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Question

Simplify the following expression.

$\frac{x^{\sqrt{2}}}{x^2}*\frac{x^{\sqrt{2}}}{x^2}$

Answer

The original expression can be rewritten as

$(\frac{x^{\sqrt{2}}}{x^2})^2$ . Whenever you can a fraction raised to a power, that power gets distributed out to the numerator and denominator. In mathematical terms, the new expression is

$\frac{x^{2}}{x^4}$ , which simply becomes $\frac{1}{x^2}$ , or $x^{-2}$

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