T1_LO6 · lo-yr9-powers-and-roots
A power (or exponent) tells you how many times to multiply a number by itself. A root asks: what number, multiplied by itself, gives the original?
Steps — evaluate powers
Identify the base (the big number) and the exponent (the small number).
Multiply the base by itself exponent times.
Example: \(2^{3
= 2 \times 2 \times 2 = 8\).}
Steps — square roots
Ask: what number times itself equals the given?
Example: \(\sqrt{25
= 5\) because $5 \times 5 = 25$.}
Key facts
$2 \times 2 \times 2 \times 2 = 2^{4}$
$2^{3} = 8$
$3^{2} = 9$
$\sqrt{25} = 5$
$4^{1} = 4$ (any number to the power 1 is itself)
$\sqrt{4} = 2$ (rational — an integer)
Thinking \(2^{3
= 6\) (multiplying base by exponent). Remember: $2^{3} = 2 \times 2 \times 2 = 8$, not $2 \times 3 = 6$.}
Write $2 \times 2 \times 2 \times 2$ using a power.
What is $2^{3}$?
What is $\sqrt{25}$?
Example 1: square numbers
Evaluate \(3^{2
\).$$ 3^{2} = 3 \times 3 = 9 $$Answer: $9$. $9$ is called a square number.}
Example 2: cube numbers
Calculate \(5^{3
\).$$ 5^{3} = 5 \times 5 \times 5 = 125 $$Answer: $125$. $125$ is called a cube number.}
Evaluate $3^{2}$.
Find $\sqrt{36}$.
Name one cube number.
Example 3: powers of 10
Calculate \(10^{3
\).$$ 10^{3} = 10 \times 10 \times 10 = 1000 $$Pattern: the exponent tells how many zeros.}
Example 4: expressing as a power
Write $81$ as a power of $3$.\[81 = 3 \times 3 \times 3 \times 3 = 3^{4
\]Answer: $3^{4}$.}
Confusing \(-3^{2