1.04 Review: Exponents and radicals
Exponents
An exponent (or power) is a small number placed in the upper right-hand corner (superscript) of another number to indicate how many times the base number is being multiplied by itself.
For example, in the expression $10^3$103 the number $10$10 is the base and the number $3$3 is the exponent (or power). The expression $10^3$103 is the same as $10\times10\times10$10×10×10, or the number $10$10 multiplied $3$3 times.
Think of the base as being closest to the ground, and the exponent (or power) is above.
We often encounter an exponent of $2$2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number $x$x with an exponent of $2$2, can be expressed as $x^2$x2, and is often read as "$x$x to the power of $2$2" (or "$x$x squared").
A number $x$x to the power of $3$3, which can be expressed as $x^3$x3, is also known as "$x$x cubed". A power of $3$3 is involved in calculations like measuring the volume of a cube.
A base number $x$x to the power of any other number $n$n, can be expressed as $x^n$xn and can be read as "$x$x to the power of $n$n".
Perfect Squares
A perfect square is the result of squaring an integer. It is useful to memorize the squares of the first 20 integers as these can save you the trouble of time consuming calculations. However, even if you cannot memorize all perfect squares, it is important to learn how to spot perfect squares.
Use the applet below to explore the first twelve square numbers.
Practice questions
Question 1
State the base for the expression $3^2$32.
question 2
Identify the power for the expression $4^6$46.
$6$6
A$4$4
B
Squaring a negative number
When we multiply two negative numbers the product is positive. This means that if we square a negative number we end up with a positive result.
$-3^2$−32 is not the same as $\left(-3\right)^2$(−3)2
$-3^2$−32 means $-\left(3^2\right)$−(32) or $-1\times\left(3\times3\right)$−1×(3×3), which gives us an answer of $-9$−9 because we are taking the square of $3$3 and then multiplying by $-1$−1.
$\left(-3\right)^2$(−3)2 means $\left(-3\right)\times\left(-3\right)$(−3)×(−3), which gives us an answer of $9$9 because the parentheses mean we are taking the square of $-3$−3.
Cubing a negative number
When we cube a negative number, we'll end up with a negative number. How does that work? Let's take a look.
Worked example
Question 3
Evaluate: $\left(-2\right)^3$(−2)3
Think: $\left(-2\right)^3=\left(-2\right)\times\left(-2\right)\times\left(-2\right)$(−2)3=(−2)×(−2)×(−2). We know multiplying two negative numbers will give us a positive answer, so $\left(-2\right)\times\left(-2\right)=4$(−2)×(−2)=4.
So, when we multiply the first two negative numbers, we get a positive answer.
Then we multiply a positive and a negative number, which gives us a negative answer.
Do:
| $\left(-2\right)^3$(−2)3 | $=$= | $\left(-2\right)\times\left(-2\right)\times\left(-2\right)$(−2)×(−2)×(−2) |
| $=$= | $4\times\left(-2\right)$4×(−2) | |
| $=$= | $-8$−8 |
Reflect: The cube of a negative number is a negative number. Can we make a generalization about even versus odd powers?
Practice questions
Question 4
$\left(-5\right)^{13}$(−5)13 simplifies to which of the following?
$5^{-13}$5−13
A$-5^{13}$−513
B$-5^{-13}$−5−13
C$5^{13}$513
D
Question 5
Evaluate $\left(-2\right)^4$(−2)4.
Using the order of operations
Now we need to consider our order of operations again. Where do exponents (or powers) fit in?
- Perform all operations within grouping symbols.
- Evaluate all exponents (or powers).
- Do all multiplications and divisions in the order in which they occur, working from left to right.
- Do all additions and subtractions in the order in which they occur, working from left to right.
Let's look at an example that put all these rules together.
Worked example
Question 6
Evaluate: $12^2-\left(-2\right)^3+27$122−(−2)3+27
Think: Calculate the square of $12$12 and the cube of $\left(-2\right)$(−2) first, and then perform the subtraction and addition working from left to right.
Do:
| $12^2-\left(-2\right)^3+27$122−(−2)3+27 | $=$= | $144-\left(-8\right)+27$144−(−8)+27 |
| $=$= | $144+8+27$144+8+27 | |
| $=$= | $179$179 |
Practice question
question 7
Evaluate $3^3-\left(-2\right)^3+59$33−(−2)3+59
Finding the square root
Finding square roots is the opposite of finding squares (square root is the inverse operation to squaring). You find the square root of a number (let's call it number A) by finding the number that, when multiplied by itself produces that number (A).
You might also see the square root symbol written with a number inside it. For example, $\sqrt{25}$√25. This means find the square root of $25$25.
Worked examples
Question 8
Evaluate: What is the square root of $144$144?
Think: What number, when squared results in $144$144? $12\times12=144$12×12=144
Do: The square root of $144$144 is $12$12
question 9
Evaluate: $\sqrt{64}$√64
Think: What number, when squared results in $64$64? $8\times8=64$8×8=64
Do: $\sqrt{64}=8$√64=8
Evaluating expressions with square roots
Now let's look at putting all this knowledge together in different types of questions.
Worked examples
question 10
Evaluate: $\sqrt{100}-\sqrt{49}$√100−√49
Think: The square root of $100$100 is $10$10 and the square root of $49$49 is $7$7.
Do:
| $\sqrt{100}-\sqrt{49}$√100−√49 | $=$= | $10-7$10−7 |
| $=$= | $3$3 |
question 11
Evaluate: $\sqrt{14+11}$√14+11
Think: Since $14+11$14+11 is all under the square root, it is like it is in imaginary parentheses and you evaluate this first.
Do:
| $\sqrt{14+11}$√14+11 | $=$= | $\sqrt{25}$√25 |
| $=$= | $5$5 |
Practice questions
question 12
Evaluate $\sqrt{8^2+6^2}$√82+62
question 13
Evaluate $\sqrt{49}+\sqrt{25}$√49+√25