Separate the numerical coefficients and the variable terms.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems on Laws of Indices:

1. Simplify the expression $\frac{30x^4}{3x^{-1}}$

Separate the numerical coefficients and the variable terms.
$\frac{30x^4}{3x^{-1}} = \left(\frac{30}{3}\right) \times \left(\frac{x^4}{x^{-1}}\right)$
Simplify the numerical part.
$\frac{30}{3} = 10$
Simplify the variable part using the rule $\frac{a^m}{a^n} = a^{m-n}$ .
$\frac{x^4}{x^{-1}} = x^{4 - (-1)} = x^{4+1} = x^5$
Combine the simplified parts.
$10 \times x^5 = 10x^5$
Answer:
The simplified expression is $\boxed{\mathbf{10x^5}}$ .

2. Simplify $\left(\frac{x^4}{2y^2}\right)^{-2}$
Apply the negative exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .
$\left(\frac{x^4}{2y^2}\right)^{-2} = \left(\frac{2y^2}{x^4}\right)^2$
Apply the power of a quotient rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
$\left(\frac{2y^2}{x^4}\right)^2 = \frac{(2y^2)^2}{(x^4)^2}$
Apply the power of a product rule $(ab)^n = a^n b^n$ and the power of a power rule $(a^m)^n = a^{mn}$ .
$\frac{(2y^2)^2}{(x^4)^2} = \frac{2^2 (y^2)^2}{x^{4 \times 2}}$ $= \frac{4y^{2 \times 2}}{x^8}$ $= \frac{4y^4}{x^8}$
Answer:
The simplified expression is $\boxed{\mathbf{\frac{4y^4}{x^8}}}$ .

3. Simplify the powers: $\frac{x^6 y^2 z^{-3}}{x^{-1} y^2 z^{-5}}$
Apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ to each variable separately.
For $x$ :
$\frac{x^6}{x^{-1}} = x^{6 - (-1)} = x^{6+1} = x^7$
For $y$ :
$\frac{y^2}{y^2} = y^{2-2} = y^0 = 1$
For $z$ :
$\frac{z^{-3}}{z^{-5}} = z^{-3 - (-5)} = z^{-3+5} = z^2$
Multiply the simplified terms together.
$x^7 \times 1 \times z^2 = x^7 z^2$
Answer:
The simplified expression is $\boxed{\mathbf{x^7 z^2}}$ .

Separate the numerical coefficients and the variable terms.

Step-by-step solution

Separate the numerical coefficients and the variable terms.

Simplify the numerical part.

Simplify the variable part using the rule $\frac{a^m}{a^n} = a^{m-n}$ .

Combine the simplified parts.

Apply the negative exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Apply the power of a quotient rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Apply the power of a product rule $(ab)^n = a^n b^n$ and the power of a power rule $(a^m)^n = a^{mn}$ .

Apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ to each variable separately.

Multiply the simplified terms together.

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Separate the numerical coefficients and the variable terms.

Step-by-step solution

Separate the numerical coefficients and the variable terms.

Simplify the numerical part.

Simplify the variable part using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Combine the simplified parts.

Apply the negative exponent rule (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n(ba​)−n=(ab​)n.

Apply the power of a quotient rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}(ba​)n=bnan​.

Apply the power of a product rule (ab)n=anbn(ab)^n = a^n b^n(ab)n=anbn and the power of a power rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Apply the division rule for exponents aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n to each variable separately.

Multiply the simplified terms together.

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Simplify the variable part using the rule $\frac{a^m}{a^n} = a^{m-n}$ .

Apply the negative exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Apply the power of a quotient rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Apply the power of a product rule $(ab)^n = a^n b^n$ and the power of a power rule $(a^m)^n = a^{mn}$ .

Apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ to each variable separately.