Dr. Algebra

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Exponents

Exponents are the little numbers on the top right of another number.  They are an easy way to show how many times a number or expression is multiplied by itself. The exponent may also be referred to as the power, or sometimes the order,  but they all have mean the same thing.  For example, if you multiply 3 times 3, you get nine.  3 times 3 times 3 gives you 27, and 3 times 3 times 3 times 3 gives you 81.  Using exponents simply gives us an easy way to write these expressions out.

    • [katex]3\cdot 3=3^{2}[/katex]
    • [katex]3\cdot 3\cdot 3=3^{3}[/katex]
    • [katex]3\cdot 3\cdot 3\cdot 3\cdot=3^{4}[/katex]

Here is an example with a variable.

[katex]x^3[/katex]

In the expression above, the number ‘x’ is the base and the number ‘3’ is the exponent.  The exponent may also be referred to as the power, or sometimes the order.

CLICK HERE for a short VIDEO Lesson

Adding and subtracting with exponents. When expressions or equations occur with terms containing exponents, they often can be simplified by combining terms with like bases and exponents as in the examples below:

    • [katex]x^{2}+x^{2}=2x^{2}[/katex]
    • [katex]4y^{3}-y^{3}=3y^{3}[/katex]
    • [katex]z^{4}y+2z^{4}y=2z^{4}y[/katex]

CLICK HERE for a short VIDEO Lesson

If the above examples are expanded (extra step shown in red), we can clearly see why and how they can be combined:

    • [katex]x^{2}+x^{2}=2xx+xx=2x^{2}[/katex]
    • [katex]4y^{3}-y^{3}=4yyy-yyy=3y^{3}[/katex]
    • [katex]z^{4}y+2z^{4}y=zzzzy+2zzzy=2z^{4}y[/katex]

Note of Caution:  It is very easy and tempting to mix up the coefficient and exponents.  This is made more tricky when the exponent is contained inside a parenthesis with an exponent outside.  This is the only time a coefficient is affected by the exponent.

    • [katex](2x)^{2}=(2)^{2}x^{2}=4x^{2}[/katex]
    • [katex](3xy)^{3}=(3)^{3}x^{3}y^{3}=27x^{3}y^{3}[/katex]
    • [katex]3z^{4}=3zzzz[/katex]

Properties of Exponents.  The properties marked with MEMORIZE should be committed to long term memory.  The two more advanced properties are less likely to be encountered but everyone should be at least familiar with them and they are provided for reference.

    • Identities  (MEMORIZE)
      • [katex]x^{0}=1[/katex]                — Anything to the zero power is always 1
      • [katex]x^{1}=x[/katex]                — Anything to the one power is always itself
    • Addition and Subtraction (MEMORIZE)
      • Must have common base
      • Must have common exponent
      • Coefficients then add and/or subtract
      • [katex]x^{a}+2x^{a}=3x^{a}[/katex]
      • [katex]4x^{a}-x^{a}=3x^{a}[/katex]
      • CLICK for Video
    • Products and Quotients (MEMORIZE)
      • [katex]x^{a}x^{b}=x^{a+b}[/katex]     — Exponents add when same base multiplied
      • [katex]\frac{x^{a}}{x^{b}}[/katex]                  –Exponents subtract when same base divided
      • CLICK for MULTIPLICATION Video
      • CLICK for DIVIDING Video
    • Exponents to a Power

      • [katex](x^{a})^{b}=x^{ab}[/katex]         — Power to another power
      • [katex](xy)^{a}=x^{a}y^{a}[/katex]        — Product to a power
    • Negative Exponents and Fractional Exponents

      • [katex]x^{-a}=\frac{1}{x^{a}}[/katex]    — Negative Power Rule
      • [katex]x^{\frac{a}{b}}=\sqrt[b]{x^{a}}[/katex]  — Fractional Power Rule

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