Simple Exponents - Algebra II

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Question

How many perfect squares satisfy the inequality $100 \leq x \leq 1,000$ ?

Answer

The smallest perfect square between 100 and 1,000 inclusive is 100 itself, since $10^{2} = 100$ . The largest can be found by noting that $\sqrt{1,000} \approx 31.6$ ; this makes $31^{2} = 961$ the greatest perfect square in this range.

Since the squares of the integers from 10 to 31 all fall in this range, this makes perfect squares.

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Question

Simplify the following expression

$3^{8}\times3^{10}$

Answer

$3^{8}\times3^{10}=3^{8+10}=3^{18}$

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Question

Simplify the following expression

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

Answer

$\frac{x^{7}}{x^{3}}=x^{7-3}=x^{4}$

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Question

Evaluate the following expression:

$\frac{2^{3}}{2^{5}}$

Answer

$\frac{2^{3}}{2^{5}}=2^{3-5}=2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}$

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Question

Solve for x:

$3^{3x+1}=81$

Answer

Solve for x:

$3^{3x+1}=81$

Step 1: Represent exponentially with a base of

$3\cdot 3\cdot 3\cdot 3=81$ , therefore

Step 2: Set the exponents equal to each other and solve for x

$3^{3x+1}=3^4$

${\color{Red} x=1}$

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Question

Solve for :

$256=2^{3x+2}$

Answer

Rewrite in exponential form with a base of :

$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=256$

Solve for by equating exponents:

$2^{3x+2}=2^{8}$

${\color{Red} x=2}$

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Question

Solve for :

$1024=4^{2x+1}$

Answer

Represent in exponential form using a base of :

$4\cdot 4\cdot 4\cdot 4\cdot 4=1024$

Solve for by equating exponents:

$4^{2x+1}=4^5$

${\color{Red} x=2}$

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Question

Simplify .

Answer

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:

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Question

Solve for :

Answer

The first step in solving for x is to simplify the right side:

$2^22^x=2^{2+x}$ .

Next, we can re-express the left side as an exponential with 2 as the base.

Now set the new left side equal to the new right side.

$2^4=2^{2+x}$

With the bases now being the same, we can simply set the exponents equal to each other.

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Question

Solve for :

$64=\frac{4^6}{4^x}$

Answer

To solve for x, we need to simplify both sides in order to make the equation simpler to solve.

can be rewritten as , and $\frac{4^6}{4^x}$ can be written as $4^{6-x}$ .

Setting the two sides equal to each other gives us

$4^3=4^{6-x}$

Since the bases are the same we can set the exponents equal to each other.

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Question

Expand:

Answer

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

Therefore:

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Question

Expand:

Answer

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

Therefore:

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Question

Evaluate:

Answer

is expanded to .

We will simply multiply the values in order to get the answer:

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Question

Evaluate:

Answer

is expanded to .

We will simply multiply the values in order to get the answer:

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Question

Expand:

$5^{-4}$

Answer

When exponents are negative, we can express them usinf the following format: $x^{-a}=\frac{1}{x^a}$

in this formula, is a positive exponent and is the base.

We can express $5^{-4}$ as $(5^{-1})^4$ , which is also the same as:

$\frac{1}{5}\frac{1}{5}\frac{1}{5}*\frac{1}{5}$

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Question

Evaluate:

$8^{-3}$

Answer

When exponents are negative, we can express them usinf the following format: $x^{-a}=\frac{1}{x^a}$

in this formula, is a positive exponent and is the base.

Therefore:

$8^{-3}=\frac{1}{8}\frac{1}{8}\frac{1}{8}=\frac{1}{512}$

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Question

Simplify:

Answer

When adding exponents, we first need to factor common terms.

Let's start by factoring out the following:

Factor.

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Question

Expand:

Answer

When a number is raised by an exponent, the base value is multiplied by itself the number of times that the exponential value indicates.

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Question

Expand:

Answer

When a number is raised by an exponent, the base value is multiplied by itself the number of times that the exponential value indicates.

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Question

Evaluate:

Answer

When a number is raised by an exponent, the base value is multiplied by itself the number of times that the exponential value indicates.

is expanded to .

The product is .

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