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1.6: Evaluate Numerical and Variable Expressions Involving Powers

Difficulty Level: Basic Created by: CK-12
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Do you know how to evaluate a numerical expression when it has powers in it? Casey is having a difficult time doing exactly that. When Casey arrived home from school he looked at this homework. He immediately noticed this problem.

\begin{align*}5^4 + -2^4 + 12\end{align*}

Casey isn't sure how to evaluate this expression. Do you know how to do it? This Concept will show you how to evaluate numerical expressions involving powers. Then you will be able to help Casey at the end of the Concept.


Did you know that you can evaluate numerical and variable expressions involving powers? First, let's identify a numerical and a variable expression.

A numerical expression is a group of numbers and operations that represent a quantity, there isn’t an equal sign.

A variable expression is a group of numbers, operations and variables that represents a quantity, there isn’t an equal sign.

We can combine the order of operations, numerical expressions and variable expressions together with powers.

Let’s talk about powers.

A power is a number with an exponent and a base. An exponent is a little number that shows the number of times a base is multiplied by itself. The base is the regular sized number that is being worked with.

Take a minute to write these definitions in your notebook.

Now let's apply this information.

\begin{align*}4^2 = 16\end{align*}

What happened here?

We can break down this problem to better understand powers and exponents. In the power \begin{align*}4^2\end{align*} or four-squared, four is the base and two is the exponent. \begin{align*}4^2\end{align*} means four multiplied two times or \begin{align*}4 \times 4\end{align*}. Therefore, \begin{align*}4^2\end{align*} is sixteen.

Let's go back a step and evaluate an expression with a power in it. Take a look.

Evaluate \begin{align*}6^3\end{align*}

First, we have to think about what this means. It means that we take the base, 6 and multiply it by itself three times.

\begin{align*}& 6 \times 6 \times 6\\ & 36 \times 6 \\ & 216\end{align*}

The answer is 216.

Here is another one with a negative number in it.

Evaluate \begin{align*}-8^2\end{align*}

To work on this one, we have to work on remembering integer rules. Think back remember that we multiply a negative times a negative to get a positive.

\begin{align*}-8 \cdot -8 = 64\end{align*}

The answer is 64.

This is called evaluating a power.

Let’s look at evaluating powers within expressions.

Simplify the expression \begin{align*}6^4 + 2^5 + 12\end{align*}.

Step 1: Simplify \begin{align*}6^4\end{align*}.

\begin{align*}6^4 = 6 \times 6 \times 6 \times 6 = 1,296\end{align*}

Step 2: Simplify \begin{align*}2^5\end{align*}.

\begin{align*}2^5 = 2 \times 2 \times 2 \times 2 \times 2= 32\end{align*}

Step 3: Add to solve.

\begin{align*}1,296 + 32 + 12 = 1,340\end{align*}

The answer is 1,340.

We can also evaluate variable expressions by substituting given values into the expressions.

Evaluate the expression \begin{align*}4a^2\end{align*} when \begin{align*}a = 3\end{align*}.

Step 1: Substitute 3 for the variable “\begin{align*}a\end{align*}.”


Step 2: Simplify the powers.

\begin{align*}& 4(3)^2\\ & 4(3 \cdot 3)\\ & 4(9)\end{align*}

Step 3: Multiply to solve.

\begin{align*}& 4(9)\\ & 36\end{align*}

The answer is 36.

Example A

\begin{align*}6^3 + 5^2 + 25\end{align*}

Solution: \begin{align*}266\end{align*}

Example B


Solution: \begin{align*}27,648\end{align*}

Example C

\begin{align*}6^2 + 5^3 + 15 - 11\end{align*}

Solution: \begin{align*}165\end{align*}

Now let's go back to the dilemma at the beginning of the Concept. Here is the problem that was puzzling to Casey.

\begin{align*}5^4 + -2^4 + 12\end{align*}

First, Casey will need to evaluate the powers.

\begin{align*}5^4 = (5)(5)(5)(5) = 625\end{align*}

\begin{align*}-2^4 = (-2)(-2)(-2)(-2) = 16\end{align*}

Now we can substitute these values back into the expression.

\begin{align*}625 + 16 + 12 = 653\end{align*}

This is the answer to Casey's problem.


Numerical Expression
a group of numbers and operations used to represent a quantity without an equals sign.
Variable Expression
a group of numbers, operations and variables used to represent a quantity without an equals sign.
the value of a base and an exponent.
the regular sized number that the exponent works upon.
the little number that tells you how many times to multiply the base by itself.

Guided Practice

Here is one for you to try on your own.

Evaluate the expression \begin{align*}5b^4 + 17\end{align*}. Let \begin{align*}b=5\end{align*}.


Step 1: Substitute 5 for “\begin{align*}b\end{align*}.”

\begin{align*}5(5)^4 + 17\end{align*}

Step 2: Simplify the powers.

\begin{align*}& 5(5 \cdot 5 \cdot 5 \cdot 5) + 17\\ & 5(625) + 17\end{align*}

Step 3: Multiply then add to solve.

\begin{align*}& 5(625) + 17\\ & 3,125 + 17 = 3,142\end{align*}

The answer is 3,142.

Video Review

[ http://www.khanacademy.org/video?v=8htcZca0JIA Khan Academy Level 1 Exponents]


Directions: Evaluate each power.

  1. \begin{align*}3^3\end{align*}
  2. \begin{align*}4^2\end{align*}
  3. \begin{align*}-2^4\end{align*}
  4. \begin{align*}-8^2\end{align*}
  5. \begin{align*}5^3\end{align*}
  6. \begin{align*}2^6\end{align*}
  7. \begin{align*}-9^2\end{align*}
  8. \begin{align*}-2^6\end{align*}

Directions: Evaluate each numerical expression.

  1. \begin{align*}6^2 + 22\end{align*}
  2. \begin{align*}-3^3 + 18\end{align*}
  3. \begin{align*}2^3 + 16 - 4\end{align*}
  4. \begin{align*}-5^2 - 19 \end{align*}
  5. \begin{align*}-7^2 + 52 - 2\end{align*}
  6. \begin{align*}18 + 9^2 - 3\end{align*}
  7. \begin{align*}22 - 3^3 + 7\end{align*}

Directions: Evaluate each variable expression using the given values.

  1. \begin{align*}6a + 4^2 - 2\end{align*}, when \begin{align*}a = 3\end{align*}
  2. \begin{align*}a^3 + 14\end{align*}, when \begin{align*}a = 6\end{align*}
  3. \begin{align*}2a^2 - 16\end{align*}, when \begin{align*}a = 4\end{align*}
  4. \begin{align*}5b^3 + 12\end{align*}, when \begin{align*}b = -2\end{align*}
  5. \begin{align*}2x^2 + 52\end{align*}, when \begin{align*}x = 4\end{align*}

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Base When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
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...
Numerical expression A numerical expression is a group of numbers and operations used to represent a quantity.
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
Power The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".
substitute In algebra, to substitute means to replace a variable or term with a specific value.
Variable Expression A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.

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