1.3: Algebra Expressions with Exponents
What if you knew the volume of a cube was represented by the formula \begin{align*} V = s^3\end{align*}, where s the length of a side. You measure the cube's side to be 4 inches. How could you find its volume? After completing this Concept, you'll be able to evaluate exponential expressions like this one.
Watch This
CK-12 Foundation: 0103S Evaluate Algebraic Expressions with Exponents
For a more detailed review of exponents and their properties, check out the video at http://www.mathvids.com/lesson/mathhelp/863-exponents---basics.
Guidance
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 we deal with integers, it is usually easiest to simplify the expression. We simplify:
\begin{align*} 2^2 &= 4\\ 2^3 &= 8\end{align*}
However, we need 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 in this case to substitute using parentheses in order to make sure that the simplification is done correctly.
Example A
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 \ \text{for} \ r.\\ & A = \pi (17)^2 \quad \pi \cdot 17 \cdot 17 \approx 907.9202 \ldots \ \text{Round to} \ 2 \ \text{decimal places.}\end{align*}
The area is approximately 907.92 square inches.
Example B
Find the value of \begin{align*}\frac { x^2y^3 } { x^3 + y^2 }\end{align*}, for \begin{align*}x = 2\end{align*} and \begin{align*}y = -4\end{align*}.
Substitute the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the following.
\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 } = \frac { - 256 } { 24 } = \frac{-32}{3} \qquad \text{Evaluate expressions:} \ (2)^2 = (2)(2) = 4 \ \text{and}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \ (2)^3 = (2)(2)(2) = 8. \ (-4)^2 = (-4)(-4) = 16 \ \text{and}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \ (-4)^3 = (-4)(-4)(-4) = -64.\end{align*}
Example C
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 \ seconds\end{align*}.
Solution
\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.
Watch this video for help with the Examples above.
CK-12 Foundation: Evaluate Algebraic Expressions with Exponents
Vocabulary
- 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).
Guided Practice
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 \text{Evaluate and simplify expressions.} \end{align*}
Practice
Evaluate 1-8 using \begin{align*}x = -1, \ y = 2, \ z = -3,\end{align*} and \begin{align*}w = 4\end{align*}.
- \begin{align*} 8x^3\end{align*}
- \begin{align*}\frac { 5x^2 } { 6z^3 } \end{align*}
- \begin{align*} 3z^2 - 5w^2\end{align*}
- \begin{align*} x^2 - y^2 \end{align*}
- \begin{align*} \frac { z^3 + w^3 } { z^3 - w^3 } \end{align*}
- \begin{align*} 2x^3 - 3x^2 + 5x - 4\end{align*}
- \begin{align*} 4w^3 + 3w^2 - w + 2 \end{align*}
- \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.
- What is the volume when \begin{align*}x = 2\end{align*}?
- What is the volume when \begin{align*}x = 3\end{align*}?
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.Image Attributions
Here you'll evaluate algebraic expressions containing exponents for specific variable values.