Powers are used when we want to multiply a number by itself repeatedly.
What are powers?
Power is the exponent that a variable is raised to. For example, the expression x² is read as x to the power of 2, or x squared, which means that the value of x is multiplied by itself as many times as the value of the power or exponent.
If the value of x is 5, then we can calculate x² like this:
\(x^2 = 5^2 = 5 \cdot 5 = 25\)
Likewise, we can calculate x3 and x4:
\(x^3 = 5^3 = 5 \cdot 5 \cdot 5 = 125\)
Notice that if you already know the value of 5², which is 25, you can multiply it by 5 one more time to obtain the value of 5³.
\(x^4 = 5^4 = 5 \cdot 5 \cdot 5 \cdot 5 = 625\)
Important to remember:
If a variable has no power or exponent, then it is assumed to be 1. For example, \(x^1 = x\)
Also, any variable to the power of 0 (zero) equals 1. For example, \(x^0 = 1\)
You can refer to Exponential Rules for a more detailed explanation of the rules you need to use when working with exponents.
Just as a reminder, these are the exponential rules that you need to keep in mind:
\(x^a \cdot x^b = x^{a+b}\)
\(\frac{x^a}{x^b} = x^{a-b}\)
\((x^a)^b = x^{a \cdot b}\)
\(x^0 = 1\)
\(x^{-a} = \frac{1}{x^a}\)
\(x^{\frac{a}{b}} = \sqrt[b]{x^a}\)
What are roots?
Roots are the inverse of powers. To calculate the nth root of a number (\(\sqrt[n]{x}\)), we need to find what number multiplied by itself n times, gives us the number inside the radical symbol (x).
1. Square root
If you want to find the square root of a number, you need to find out what number times itself would give us the number inside the square root.
If you want to find the square root of 25, you need to find what number multiplied by itself equals 25.
\(\sqrt{25} = \pm 5\)
But why is the result ± 5?
This is because the square root of 25 can be either 5 or -5.
\(5 \cdot 5 = 25\)
\((-5) \cdot (-5) = 25\)
Therefore, there are always two answers when we take the square root of a number.
\(\sqrt{-25} ≠ -5\)
The square root of a negative number has no real solution; imaginary numbers are required in this case. Only positive numbers can have their square root taken in this way.
Square roots can be classified according to the type of number inside the root as follows:
The square root of perfect squares gives an integer as a result. It is very easy to calculate, and useful to remember when working with expressions containing powers and roots. It helps to evaluate and simplify these types of expressions. Just as a reminder, here are the first ten:
| \(\sqrt{1}\) | \(\sqrt{4}\) | \(\sqrt{9}\) | \(\sqrt{16}\) | \(\sqrt{25}\) | \(\sqrt{36}\) | \(\sqrt{49}\) | \(\sqrt{64}\) | \(\sqrt{81}\) | \(\sqrt{100}\) |
| ± 1 | ± 2 | ± 3 | ± 4 | ± 5 | ± 6 | ± 7 | ± 8 | ± 9 | ± 10 |
The square root of numbers that are not perfect squares is not an integer. They produce irrational numbers with infinite decimals. To represent this type of number more exactly, they are left in their root form and called surds. For example: \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}\).
If the number inside the root of a surd has a square number as a factor, then it can be simplified. For example: \(\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}\).
You can read about Surds for more detail.
2. Cube Root
If you want to find the cube root of a number, you need to find out what number multiplied by itself 3 times would give us the number inside the cube root. It is the opposite of raising a number to the 3rd power.
If you want to find the cube root of 8, you need to find what number multiplied by itself 3 times equals 8.
\(\sqrt[3]{8} = 2\)
Notice that in this case, we have only one answer, not two. This is because when you multiply a negative number by itself 3 times, the result is also negative.
\((-2) \cdot (-2)\cdot (-2) = -8\)
Therefore, the only possible answer is:
\(2 \cdot 2 \cdot 2 = 8\)
\(\sqrt[3]{8} = -2\)
Cube roots CAN take the cube root of a negative number.
3. Other roots
How do you write powers as roots and roots as powers?
To write powers as roots and roots as powers, we need to understand how fractional exponents work.
Fractional exponents
Fractional exponents are equivalent to roots as shown in the following exponential rule:
\(x^{\frac{a}{b}} = \sqrt[b]{x^a}\)
Using this expression, you can write any fractional exponent as a root.
\(x^{\frac{1}{2}} = \sqrt{x}\)
\(x^{\frac{1}{3}} = \sqrt[3]{x}\)
\(x^{\frac{2}{3}} = \sqrt[3]{x^2}\)
You can use the same expression to write any root as a fractional exponent.
\(\sqrt[4]{x} = x^{\frac{1}{4}}\)
\(\sqrt[6]{x^5} = x^{\frac{5}{6}}\)
Evaluating and simplifying expressions with powers and roots
Now that you know how to work with fractional exponents and, keeping in mind the exponential rules, you have everything you need to evaluate or simplify expressions containing powers and roots. Here are some examples:
Example 1
Evaluate or simplify \(\sqrt{50}\)
Remembering perfect squares, you can change \(\sqrt{50}\) to \(\sqrt{25 \cdot 2}\)
\(\sqrt{50} = \sqrt{25} \cdot \sqrt{2}\)
\(\sqrt{50} = 5\sqrt{2}\)
\(5\sqrt{2}\) is an example of a surd because it cannot be simplified further, so it is left in its square root form. Remember to read more about Surds!
Example 2
Evaluate or simplify \(\frac{\sqrt{x} \sqrt[4]{x}}{\sqrt[3]{x}}\)
\(\begin{align} \frac{\sqrt{x}\sqrt[4]{x}}{\sqrt[3]{x}} &= \frac{x^{\frac{1}{2}} \cdot x^{\frac{1}{4}}}{x^{\frac{1}{3}}} \\ &= \frac{x^{\frac{3}{4}}}{x^{\frac{1}{3}}} \\ &= x^{\frac{3}{4} - \frac{1}{3}} \\ &= x^{\frac{5}{12}} \end{align}\)
Written in text format, the process goes as follows:
- Transform the roots into fractional exponents
- Use the exponential rule \(x^a \cdot x^b = x^{a+b}\)
- Use the exponential rule \(\frac{x^a}{x^b} = x^{a-b}\)
- Solve the resulting equation
Example 3
Evaluate or simplify \(\frac{24x^4y^5}{4x^5}\)
\(\frac{24x^4y^5}{4x^5} = 6x^{-1}y^5\) using the exponential rule \(\frac{x^a}{x^b} = x^{a-b}
\(\frac{24x^4y^5}{4x^5} = \frac{6y^5}{x}\) using the exponential rule \(x^{-a} = \frac{1}{x^a}\)
Example 4
Evaluate or simplify \(\big( \frac{3xy^2}{2x^3} \big)^{-2}\)
\(\begin{align} \big( \frac{3xy^2}{2x^3} \big)^{-2} &= \big(\frac{2x^3}{3xy^2} \big) \\ &= \frac{(2x^3)^2}{(3xy^2)^2} \\ &= \frac{4x^6}{9x^2y^4} \\ &= \frac{4x^4}{9y^4} \end{align}\)
In writing, the steps are:
- Use the exponential rule flip the fraction \(x^{-a} = \frac{1}{x^a}\)
- Distribute the exponent into the numerator and denominator
- Use the exponential rule \(\frac{x^a}{x^b} = x^{a-b}\)
Powers and Roots - Key takeaways
Power is the exponent that a variable or number is being raised to.
The root is the opposite of power.
Odd roots will have one solution, while even roots will have two.
Only positive numbers can have their square roots taken, without using imaginary numbers.
Negative numbers can have their cube roots taken.
Knowing the square roots of perfect squares and the exponential rules is very useful when evaluating or simplifying algebraic expressions containing powers and roots.
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