High School

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For problems 3-8, write an equivalent expression with as few factors as possible. Show your work.

3. [tex]\(2 \cdot 5 \cdot 10n\)[/tex]

[tex]\[
\begin{array}{r}
2 \cdot 5 = 10 \\
10 \cdot 10 = 100n
\end{array}
\][/tex]

5. [tex]\(8x \cdot 4y \cdot \frac{1}{2}\)[/tex]

[tex]\[
\begin{array}{l}
8x \cdot 4y = 32xy \\
32xy \cdot \frac{1}{2} = 16xy
\end{array}
\][/tex]

6. [tex]\(\frac{1}{4}g \cdot 16\)[/tex]

[tex]\[
\begin{array}{l}
\frac{1}{4} \cdot 16 = 4 \\
4g
\end{array}
\][/tex]

7. [tex]\(p \cdot \frac{7}{6} \cdot 12\)[/tex]

[tex]\[
\begin{array}{l}
\frac{7}{6} \cdot 12 = 14 \\
14p
\end{array}
\][/tex]

8. [tex]\(\frac{3}{4} \cdot \frac{5}{6} \cdot 8f\)[/tex]

[tex]\[
\begin{array}{l}
\frac{3}{4} \cdot \frac{5}{6} = \frac{15}{24} = \frac{5}{8} \\
\frac{5}{8} \cdot 8f = 5f
\end{array}
\][/tex]

9. [tex]\(6 \cdot x \cdot \frac{3}{4} \cdot \frac{5}{6}\)[/tex]

[tex]\[
\begin{array}{l}
6 \cdot \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \\
\frac{9}{2} \cdot \frac{5}{6} = \frac{45}{12} = \frac{15}{4} \\
\frac{15}{4}x
\end{array}
\][/tex]

Answer :

Sure! Let's simplify each expression step-by-step to write an equivalent expression with as few factors as possible.

### Problem 3:
[tex]\[ 2 \cdot 5 \cdot 10n \][/tex]

1. Multiply [tex]\( 2 \cdot 5 = 10 \)[/tex]
2. Now multiply this result by 10:
[tex]\[ 10 \cdot 10 = 100 \][/tex]
3. So the expression becomes:
[tex]\[ 100n \][/tex]

### Problem 5:
[tex]\[ 8x \cdot 4y \cdot \frac{1}{2} \][/tex]

1. Multiply [tex]\( 8 \cdot \frac{1}{2} = 4 \)[/tex]
2. Now multiply this result by [tex]\( 4y \)[/tex]:
[tex]\[ 4 \cdot 4y = 16y \][/tex]
3. So the expression becomes:
[tex]\[ 16xy \][/tex]

### Problem 6:
[tex]\[ p \cdot \frac{7}{6} \cdot 12 \][/tex]

1. Multiply the fractions part:
[tex]\[ \frac{7}{6} \cdot 12 = \frac{7 \cdot 12}{6} = 14 \][/tex]
2. Now multiply this result by [tex]\( p \)[/tex]:
[tex]\[ 14 \cdot p = 14p \][/tex]
3. So the expression becomes:
[tex]\[ 14p \][/tex]

### Problem 7:
[tex]\[ \frac{3}{4} \cdot \frac{5}{6} \cdot 8f \][/tex]

1. Multiply the fractions:
[tex]\[ \frac{3}{4} \cdot \frac{5}{6} = \frac{3 \cdot 5}{4 \cdot 6} = \frac{15}{24} = \frac{5}{8} \][/tex]
2. Now multiply this result by [tex]\( 8f \)[/tex]:
[tex]\[ \frac{5}{8} \cdot 8f = 5f \][/tex]
3. So the expression becomes:
[tex]\[ 5f \][/tex]

### Problem 8:
[tex]\[ 6 \cdot x \cdot \frac{3}{4} \cdot \frac{5}{6} \][/tex]

1. Combine the constants and fractions:
[tex]\[ 6 \cdot \frac{3}{4} = \frac{18}{4} = 4.5 \][/tex]
2. Then multiply by the remaining fraction:
[tex]\[ 4.5 \cdot \frac{5}{6} = \frac{4.5 \cdot 5}{6} = \frac{22.5}{6} = 3.75 \][/tex]
3. Now, include [tex]\( x \)[/tex]:
[tex]\[ 3.75x \][/tex]
4. So the expression becomes:
[tex]\[ \frac{15}{4}x \][/tex]

To summarize the simplified expressions:
1. [tex]\( 100n \)[/tex]
2. [tex]\( 16xy \)[/tex]
3. [tex]\( 14p \)[/tex]
4. [tex]\( 5f \)[/tex]
5. [tex]\( \frac{15}{4}x \)[/tex]

I hope this helps! Let me know if you have any more questions.

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Rewritten by : Barada