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# Algebra Expressions with Exponents

## Calculate values of numbers with exponents

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Algebra Expressions with Exponents

### Exponents

Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for repeated multiplication. For example:

\begin{align} 2 \cdot 2 & = 2^2\\ 2 \cdot 2 \cdot 2 & = 2^3\\ \end{align}

The exponent stands for how many times the number is used as a factor (multiplied). When dealing with integers, it's usually easiest to simplify the expression. For example, as shown here, we commonly just write 4 instead of \begin{align*}2^2,\end{align*} and 8 instead of \begin{align*}2^3.\end{align*}

\begin{align} 2^2 & = 4\\ 2^3 & = 8\\ \end{align}

However, we generally keep the exponents when we work with variables, because it is much easier to write \begin{align*}x^8\end{align*} than \begin{align*}x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x.\end{align*}

To evaluate expressions with exponents, substitute the values you are given for each variable and simplify. It is especially important when working with exponents to substitute using parentheses in order to make sure that the simplification is done correctly.

#### Evaluating an Expression Containing Exponents

The area of a circle is given by the formula \begin{align*}A = \pi r^2.\end{align*} Find the area of a circle with radius \begin{align*}r = 17 \ inches.\end{align*}

Substitute values into the equation.

\begin{align} A & = \pi r^2 \qquad \text{Substitute 17 for }r.\\ & = \pi (17)^2\\ & = 3.14 (17)(17)\\ A & \approx 907.92 \text{ Rounded to 2 decimal places.}\\ \end{align}

The area of the circle is approximately 907.92 square inches.

#### Evaluating a Multi-Variable Expression Containing Exponents

\begin{align*}\text{Find the value of }\frac { x^2y^3 } { x^3 + y^2 }, \text{ when }x=2 \text{ and }y=-4.\end{align*}

Substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align} \frac { x^2y^3 } { x^3 + y^2 } & = \frac { (2)^2 (-4)^3 } { (2)^3 + (-4)^2 } \qquad \text{Substitute} \ 2 \ \text{for} \ x \ \text{and} \ -4 \ \text{for} \ y.\\ & = \frac { 4(-64) } { 8 + 16 } \qquad \qquad (2)^2 = 4, \ (-4)^3 = -64, \ (2)^3 = 8, \ (-4)^2 = 16.\\ & = \frac { - 256 } { 24 }\\ & = \frac{-32}{3} \\ \end{align}

#### Real World Application

The height \begin{align*}(h)\end{align*} of a ball in flight is given by the formula \begin{align*}h = - 32t^2 + 60t + 20\end{align*}, where the height is given in feet and the time \begin{align*}(t)\end{align*} is given in seconds. Find the height of the ball at time \begin{align*}t = 2 \text{ seconds.}\end{align*}

\begin{align} h & = -32t^2 + 60t + 20\\ & = -32(2)^2 + 60(2) + 20 \qquad \text{Substitute} \ 2 \ \text{for} \ t.\\ & = -32(4) + 60(2) + 20\\ & = 12\\ \end{align}

The height of the ball is 12 feet.

### Example

#### Example 1

Find the value of \begin{align*}\frac { a^2+b^2 } { a^2-b^2 }\end{align*}, for \begin{align*}a = -1\end{align*} and \begin{align*}5\end{align*}.

Substitute the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the following.

\begin{align*} \frac { a^2+b^2 } { a^2-b^2 } = \frac { (-1)^2+(5)^2 } { (-1)^2-(5)^2 } \qquad \text{Substitute} \ -1 \ \text{for} \ a \ \text{and} \ 5 \ \text{for} \ b.\\ \frac { 1+25 } { 1-25 } = \frac { 26 } { 24 }=\frac{13}{12} \qquad \qquad \text{Evaluate and simplify expressions.} \end{align*}

### Review

Evaluate 1-8 using \begin{align*}x = -1, \ y = 2, \ z = -3,\end{align*} and \begin{align*}w = 4\end{align*}.

1. \begin{align*} 8x^3\end{align*}
2. \begin{align*}\frac { 5x^2 } { 6z^3 } \end{align*}
3. \begin{align*} 3z^2 - 5w^2\end{align*}
4. \begin{align*} x^2 - y^2 \end{align*}
5. \begin{align*} \frac { z^3 + w^3 } { z^3 - w^3 } \end{align*}
6. \begin{align*} 2x^3 - 3x^2 + 5x - 4\end{align*}
7. \begin{align*} 4w^3 + 3w^2 - w + 2 \end{align*}
8. \begin{align*}3 + \frac{ 1 } { z^2 }\end{align*}

For 9-10, use the fact that the volume of a box without a lid is given by the formula \begin{align*} V = 4x(10 - x)^2\end{align*}, where \begin{align*}x\end{align*} is a length in inches and \begin{align*}V\end{align*} is the volume in cubic inches.

1. What is the volume when \begin{align*}x = 2\end{align*}?
2. What is the volume when \begin{align*}x = 3\end{align*}?

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### Vocabulary Language: English

TermDefinition
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
substitute In algebra, to substitute means to replace a variable or term with a specific value.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.