Worksheet - Indices | PDF - Free Printable

You've uploaded an image containing algebra problems. I’ll solve all parts of the problems shown — specifically, m), n), and 3a) through 3d) — and explain each step clearly.

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## Problem m)
> Simplify:
> $-4pt^4 \times 6pt$

Step-by-step:

Multiply the coefficients:
$-4 \times 6 = -24$

Multiply the variables:
$p \times p = p^2$
$t^4 \times t = t^{4+1} = t^5$

So, putting it all together:
Answer: $\boxed{-24p^2t^5}$

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## Problem n)
> Simplify:
> $8x^4y^2 \times 3xy^5$

Step-by-step:

Coefficients:
$8 \times 3 = 24$

Variables:
$x^4 \times x = x^{4+1} = x^5$
$y^2 \times y^5 = y^{2+5} = y^7$

So, combining:
Answer: $\boxed{24x^5y^7}$

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## Problem 3a)
> Simplify fully:
> $\frac{x^4y^5}{x^3yz}$

Step-by-step:

Use exponent rules: $\frac{a^m}{a^n} = a^{m-n}$

For $x$:
$\frac{x^4}{x^3} = x^{4-3} = x^1 = x$

For $y$:
$\frac{y^5}{y} = y^{5-1} = y^4$

The $z$ in the denominator has no matching term in numerator → remains as $z$ in denominator.

So, result:
Answer: $\boxed{\frac{xy^4}{z}}$

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## Problem 3b)
> Simplify fully:
> $\frac{a^2b^4}{a^4b}$

Step-by-step:

For $a$:
$\frac{a^2}{a^4} = a^{2-4} = a^{-2} = \frac{1}{a^2}$

For $b$:
$\frac{b^4}{b} = b^{4-1} = b^3$

Combine:
$\frac{b^3}{a^2}$

Answer: $\boxed{\frac{b^3}{a^2}}$

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## Problem 3c)
> Simplify fully:
> $\frac{18m^3n}{6m^2n}$

Step-by-step:

First, simplify coefficients:
$\frac{18}{6} = 3$

For $m$:
$\frac{m^3}{m^2} = m^{3-2} = m$

For $n$:
$\frac{n}{n} = 1$ (they cancel out)

So, result:
Answer: $\boxed{3m}$

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## Problem 3d)
> Simplify fully:
> $\frac{20f^7h}{4f^4h^5}$

Step-by-step:

Coefficients:
$\frac{20}{4} = 5$

For $f$:
$\frac{f^7}{f^4} = f^{7-4} = f^3$

For $h$:
$\frac{h}{h^5} = h^{1-5} = h^{-4} = \frac{1}{h^4}$

Combine:
$5f^3 \cdot \frac{1}{h^4} = \frac{5f^3}{h^4}$

Answer: $\boxed{\frac{5f^3}{h^4}}$

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## ✔ Final Answers:

m) $\boxed{-24p^2t^5}$
n) $\boxed{24x^5y^7}$
3a) $\boxed{\frac{xy^4}{z}}$
3b) $\boxed{\frac{b^3}{a^2}}$
3c) $\boxed{3m}$
3d) $\boxed{\frac{5f^3}{h^4}}$

Let me know if you’d like to see these explained with visual diagrams or further examples!