1. The product of powers with the same base
Exercises
Definitions
In the expression [[$ 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16 $]], the number 2 is called the base number, the number 4 is called the exponent and the number 16 is called the value of the power. If brackets are not used, the exponent only affects the number directly below the exponent.
The base number is [[$2$]]. The answer is negative because there is an odd number (1) of negative factors in the product.
b) [[$ (-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = 16 $]]
The base number is [$-2$]]. The answer is positive because there is an even number (4) of negatives factors in the product.
Example 1
Simplify the powers.
a) [[$ -2^4 = -2 \cdot 2 \cdot 2 \cdot 2 = -16 $]]The base number is [[$2$]]. The answer is negative because there is an odd number (1) of negative factors in the product.
b) [[$ (-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = 16 $]]
The base number is [$-2$]]. The answer is positive because there is an even number (4) of negatives factors in the product.
The product of powers with the same base
In the product [[$ 4^2 \cdot 4^3 $]], the base number of both powers is the same. The expression is called the product of powers with the same base.
[[$ 4^2 \cdot 4^3 = \underbrace{4 \cdot 4}_{2 \text{ kpl}} \cdot \underbrace{4 \cdot 4 \cdot 4}_{+3 \text{ kpl}} = 4^5 = 1024 $]]
[[$ 4^2 \cdot 4^3 = \underbrace{4 \cdot 4}_{2 \text{ kpl}} \cdot \underbrace{4 \cdot 4 \cdot 4}_{+3 \text{ kpl}} = 4^5 = 1024 $]]
The product of powers with the same base.
The exponents of similar powers are multiplied by one another by adding them together. The base number stays the same.
[[$ a^m \cdot a^n = a^{m+n} $]]
Example 1
Simplify the powers.| a) | [[$ a^2 \cdot a^4 = a^{2+4} = a^6 $]] | |
| b) | [[$ x \cdot x^2 \cdot x^3 = x^{1+2+3} = x^6 $]] | |
| c) | [[$ a^3 \cdot a^2 \cdot b \cdot b^6= a^{3+2} \cdot b^{1+6} = a^5b^7 $]] |
Only powers powers with the same base can be combined. |
Multiplications of powers with the same base often involve other factors that can be combined separately. If there are variables or letters in the product, the multiplication sign is omitted between the numeric value and the variable.
Example 2
Simplify the powers.
| a) | [[$ (-2) \cdot (-2)^2 = (-2)^{1+2} = (-2)^3 = -8 $]] | |
| b) | [[$ -3 \cdot 3^3 = -3^{1+3} = -3^4 = -81 $]] | |
| c) | [[$ 2x^2 \cdot x^6 = 2x^{2+6} = 2x^8 $]] | |
| d) | [[$ 3a^4 \cdot (-2a^3) = 3 \cdot (-2) \cdot a^4 \cdot a^3 = -6a^{4+3} = -6a^7 $]] |
The numbers are multiplied and the exponents are added together. |