Module 15: Background You’ll Need 5

  • Simplify exponents

exponents

The mathematical operation of exponentiation is denoted using superscript notation:

[latex]b^x[/latex]

Some ways that this can be read include “[latex]b[/latex] raised to the power of [latex]x[/latex]” or “[latex]b[/latex] raised to the [latex]x[/latex] power.”

The quantity [latex]b[/latex] is called the base, and the quantity [latex]x[/latex] is called the exponent.

When the exponent is a positive integer, the exponent describes how many times to multiply the base by itself.

When the exponent is 2, we say that we are squaring the base. The quantity

[latex]b^2[/latex]

can be read as “[latex]b[/latex] raised to the power of [latex]2[/latex]” or “[latex]b[/latex] raised to the second power,” as described above, but it can also be read as “[latex]b[/latex] squared.”

For example: [latex]4^2 = 4 \cdot 4 = 16[/latex]

When the exponent is 3, we say that we are cubing the base. The quantity

[latex]b^3[/latex]

can be read as “[latex]b[/latex] raised to the power of [latex]3[/latex]” or “[latex]b[/latex] raised to the third power,” as described above, but it can also be read as “[latex]b[/latex] cubed.”

For example: [latex]4^3 = 4 \cdot 4 \cdot 4 = 64[/latex]

Calculate [latex]2^4[/latex].
Notice that the base is [latex]2[/latex] and the exponent is [latex]4[/latex].
[latex]4[/latex] describes how many times to multiply the base [latex]2[/latex] by itself.[latex]2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16[/latex]
What you have discovered is that when an exponent is negative, it tells us to take the reciprocal of the result we get when we have a positive exponent. In other words,

[latex]b^{-x} = \dfrac{1}{b^x}[/latex]