How to Multiply Numbers Raised to Exponents

By Barbara Conn
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Walking through a potential apartment, house or office space, you use exponents, perhaps subconsciously, whenever you consider total and component area sizes. One square foot, for example, is 1 foot by 1 foot, or 1 foot raised to the exponent (or power of) 2, or (1 foot)^2. You also use exponents when specifying computer RAM in gigabytes (10^9 bytes) or external hard drive storage capacity in terabytes (10^12 bytes).

Definitions

Step 1

Recognize an exponent and its base. An exponent indicates the number of times to multiply a number (or base) by itself.

Examples: The number 4 raised to the power of (or exponent) 3 = 4^3 = 4 x 4 x 4 = 64. 12^5 = 12 x 12 x 12 x 12 x 12 = 248,832.

Step 2

Learn the basic rule for the exponent 0 (zero). Any number (with the possible exception of 0) raised to the exponent (or power) 0 equals 1.

Examples: 1^0 = 1.

10^0 = 1.

673^0 = 1.

Step 3

Learn the meaning of a negative exponent. The top number in a simple fraction displayed in stacked format (or, in single-line format, the expression preceding the slash, “/”) is the numerator. The bottom number is the denominator. For example, in the fraction 1/2, the numerator is 1 and the denominator is 2. When a whole number is raised to a power containing a negative number, it becomes a fraction. The value of the numerator equals 1, and value of the denominator equals the whole number raised to the power without the minus sign. When a fraction is raised to a power containing a negative number, the expressions in the numerator and denominator switch positions.

Examples: 5^-3 = 1/5^3 = 1/(5 x 5 x 5) = 1/125.

(1/4)^-3 = 4^3 = 4 x 4 x 4 = 64.

(4/5)^-1 = 5/4 = 1.25.

(4/5)^-3 = (5 x 5 x 5)/(4 x 4 x 4) = 125/64 = 1.95 (with rounding to 3 digits).

Basic Rules, Shortcuts and Exercises

Step 1

Multiply by a number with the exponent 0. Any number with the exponent 0 equals 1.

Example: 4^0 x 4^3 = 1 x (4 x 4 x 4) = 1 x 64 = 64.

Step 2

Multiply by another number.

Example: 10^3 x 4.3 = (10 x10 x 10) x 4.3 = 1,000 x 4.3 = 4,300.

Step 3

Multiply a number raised to a negative power by another number.

Example: 10^-3 x 4.3 = 1/(10 x 10 x 10) x 4.3 = 1/1000 x 4.3 = 0.0043.

Step 4

Multiply two numbers with the same base and different exponents.

A shortcut is to add the exponents.

Example: 4^3 x 4^6 = 4^(3 + 6) = 4^9 = 262,144.

Step 5

Calculate the value of a number with an exponent raised to a power.

A shortcut is to multiply the exponents.

Example: (3^2)^4 = 3^(2 X 4) = 3^8 = 6,561.

Step 6

Multiply two different numbers raised to different powers.

Example: 3^4 x 6^3 = (3 x 3 x 3 x 3) x (6 x 6 x 6) = 81 x 216 = 17,496.

References

About the Author

Based in Washington, DC, award-winning editor Barbara Conn has been writing about science, technology, small business, and general interest topics since 1984. Her articles have appeared in the Capital PC User Group “Monitor.” She earned a Bachelor of Science degree in chemistry from Bucknell University.