Pre-Algebra Examples

 $\begin{array}{r} h = & 7.5 \\ l = & 6 \\ w = & 9 \end{array}$

Step 1

The surface area of a pyramid is equal to the sum of the areas of each side of the pyramid. The base of the pyramid has area $l w$ , and $s_{l}$ and $s_{w}$ represent the slant height on the length and slant height on the width.

$(l e n g t h) \cdot (w i d t h) + (w i d t h) \cdot s_{l} + (l e n g t h) \cdot s_{w}$

Step 2

Substitute the values of the length $l = 6$ , the width $w = 9$ , and the height $h = 7.5$ into the formula for surface area of a pyramid.

$6 \cdot 9 + 9 \cdot \sqrt{{(\frac{6}{2})}^{2} + {(7.5)}^{2}} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $6$ by $9$ .

$54 + 9 \cdot \sqrt{{(\frac{6}{2})}^{2} + {(7.5)}^{2}} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3.2

Divide $6$ by $2$ .

$54 + 9 \cdot \sqrt{3^{2} + {(7.5)}^{2}} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3.3

Raise $3$ to the power of $2$ .

$54 + 9 \cdot \sqrt{9 + {(7.5)}^{2}} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3.4

Raise $7.5$ to the power of $2$ .

$54 + 9 \cdot \sqrt{9 + 56.25} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3.5

Add $9$ and $56.25$ .

$54 + 9 \cdot \sqrt{65.25} + 6 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(7.5)}^{2}}$

Step 3.6

Apply the product rule to $\frac{9}{2}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{9^{2}}{2^{2}} + {(7.5)}^{2}}$

Step 3.7

Raise $9$ to the power of $2$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81}{2^{2}} + {(7.5)}^{2}}$

Step 3.8

Raise $2$ to the power of $2$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81}{4} + {(7.5)}^{2}}$

Step 3.9

Raise $7.5$ to the power of $2$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81}{4} + 56.25}$

Step 3.10

To write $56.25$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81}{4} + 56.25 \cdot \frac{4}{4}}$

Step 3.11

Combine $56.25$ and $\frac{4}{4}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81}{4} + \frac{56.25 \cdot 4}{4}}$

Step 3.12

Combine the numerators over the common denominator.

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81 + 56.25 \cdot 4}{4}}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $56.25$ by $4$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{81 + 225}{4}}$

Step 3.13.2

Add $81$ and $225$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{306}{4}}$

Step 3.14

Cancel the common factor of $306$ and $4$ .

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Step 3.14.1

Factor $2$ out of $306$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{2 (153)}{4}}$

Step 3.14.2

Cancel the common factors.

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Step 3.14.2.1

Factor $2$ out of $4$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{2 \cdot 153}{2 \cdot 2}}$

Step 3.14.2.2

Cancel the common factor.

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{2 \cdot 153}{2 \cdot 2}}$

Step 3.14.2.3

Rewrite the expression.

$54 + 9 \sqrt{65.25} + 6 \cdot \sqrt{\frac{153}{2}}$

Step 3.15

Rewrite $\sqrt{\frac{153}{2}}$ as $\frac{\sqrt{153}}{\sqrt{2}}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{\sqrt{153}}{\sqrt{2}}$

Step 3.16

Simplify the numerator.

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Step 3.16.1

Rewrite $153$ as $3^{2} \cdot 17$ .

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Step 3.16.1.1

Factor $9$ out of $153$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{\sqrt{9 (17)}}{\sqrt{2}}$

Step 3.16.1.2

Rewrite $9$ as $3^{2}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{\sqrt{3^{2} \cdot 17}}{\sqrt{2}}$

Step 3.16.2

Pull terms out from under the radical.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17}}{\sqrt{2}}$

Step 3.17

Multiply $\frac{3 \sqrt{17}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot (\frac{3 \sqrt{17}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.18

Combine and simplify the denominator.

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Step 3.18.1

Multiply $\frac{3 \sqrt{17}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.18.2

Raise $\sqrt{2}$ to the power of $1$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.18.3

Raise $\sqrt{2}$ to the power of $1$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.18.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.18.5

Add $1$ and $1$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.18.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.18.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.18.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.18.6.3

Combine $\frac{1}{2}$ and $2$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.18.6.4

Cancel the common factor of $2$ .

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Step 3.18.6.4.1

Cancel the common factor.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.18.6.4.2

Rewrite the expression.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{2^{1}}$

Step 3.18.6.5

Evaluate the exponent.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{17} \sqrt{2}}{2}$

Step 3.19

Simplify the numerator.

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Step 3.19.1

Combine using the product rule for radicals.

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{2 \cdot 17}}{2}$

Step 3.19.2

Multiply $2$ by $17$ .

$54 + 9 \sqrt{65.25} + 6 \cdot \frac{3 \sqrt{34}}{2}$

Step 3.20

Cancel the common factor of $2$ .

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Step 3.20.1

Factor $2$ out of $6$ .

$54 + 9 \sqrt{65.25} + 2 (3) \cdot \frac{3 \sqrt{34}}{2}$

Step 3.20.2

Cancel the common factor.

$54 + 9 \sqrt{65.25} + 2 \cdot 3 \cdot \frac{3 \sqrt{34}}{2}$

Step 3.20.3

Rewrite the expression.

$54 + 9 \sqrt{65.25} + 3 \cdot (3 \sqrt{34})$

Step 3.21

Multiply $3$ by $3$ .

$54 + 9 \sqrt{65.25} + 9 \sqrt{34}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$179.1783$