Introduction

To evaluate a large exponent, we first write out the power using multiplication signs as usual. Then, we calculate the product by splitting it into groups of two-digit multiplications, before calculating the resulting product.

For example, let's evaluate 2^4.

First, we write 2^4 = \underbrace{2 \times 2 \times 2 \times 2}_{\large\text{[math]4[/math] times}}.

Since 2 \times 2 = 4, we have \underbrace{2 \times 2}_{\large \color{blue}4} \times \underbrace{2 \times 2}_{\large \color{blue}4} = {\color{blue}4} \times {\color{blue}4}.

Finally, we work out 4\times 4. Since 4\times 4=16, we conclude that 2^4=16.

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Example: Squaring Two-Digit Numbers

Evaluate 31^2.

EXPLANATION

To evaluate 31^2, we start by writing it out with a multiplication sign: 31^2 = \underbrace{31 \times 31}_{\large\text{[math]2[/math] times}}

Then, we work out 31 \times 31.

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} %31 + 930 = {\color{red} 961} \end{array} \\[5pt] & \end{align*}

Therefore, 31 \times 31 = 961.

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Practice: Squaring Two-Digit Numbers

$15^2=$

a
 $225$
b
 $125$
c
 $200$
d
 $355$
e
 $30$
Practice: Squaring Two-Digit Numbers

$26^2 =$

a
b
c
d
e
Practice: Squaring Two-Digit Numbers

$76^2=$

a
 $ 5,566$
b
 $ 576$
c
 $ 5,776$
d
 $ 5,666$
e
 $ 5,956$
Example: Squaring Numbers Ending in Zeros

Evaluate 220^2.

EXPLANATION

To evaluate 220^2, we start by writing it out with a multiplication sign: 220^2 = \underbrace{220 \times 220}_{\large\text{[math]2[/math] times}}

Then, we work out 220 \times 220.

First, we figure out the non-zero part and the block of zeros for both numbers: \begin{align*} 220 &\quad\longrightarrow\quad \bbox[2px, lightgray]{22} \color{blue}{0}\\[2pt] 220 &\quad\longrightarrow\quad \bbox[2px, lightgray]{22} \color{blue}{0} \end{align*}

We can see that there are 2 zeros in total.

Next, we multiply the non-zero parts:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} %125 + 500 = {\color{red} 625} \end{array} \\[5pt] & \end{align*}

Finally, we join the above with the block of zeros: {\color{red}484}{\color{blue}00}= 48,400

Therefore, 220^2 = 48,400.

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Practice: Squaring Numbers Ending in Zeros

$60^2=$

a
b
c
d
e
Practice: Squaring Numbers Ending in Zeros

$200^2=$

a
 $4,000$
b
 $400$
c
 $8,000$
d
 $20,000$
e
 $40,000$
Practice: Squaring Numbers Ending in Zeros

$150^2=$

a
b
c
d
e
Example: Raising Numbers to Larger Exponents

Evaluate 7^4.

EXPLANATION

According to the definition, 7^4 = \underbrace{7 \times 7 \times 7 \times 7}_{\large\text{[math]4[/math] times}}.

Since 7 \times 7 = 49, we have \underbrace{7 \times 7}_{\color{blue}49} \times \underbrace{7 \times 7}_{\color{blue}49} = {\color{blue}49} \times {\color{blue}49}.

Finally, we work out 49 \times 49.

\begin{align*} \begin{array}{ccccc} & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}3[/math]} \\[2pt] \fbox{[math]\color{blue}8[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!9\!\!\!\! \\ \hline & & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \end{array}\end{align*}

Therefore, 7^4=2,401.

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Practice: Raising Numbers to Larger Exponents

$3^4=$

a
 $3,333$
b
 $12$
c
 $333$
d
 $7$
e
 $81$
Practice: Raising Numbers to Larger Exponents

$4^4=$

a
b
c
d
e
Practice: Raising Numbers to Larger Exponents

Given that $17^2 = 289,$ evaluate $17^4.$

a
b
c
d
e
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