Simplify the following algebraic expressions: 1. (30x4)/(3x⁻²) 2. (x3/22)⁻² 3. x6y2z−3x−1z⁻⁵

Question

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Here are the solutions to the questions:

1. Simplify the expression $\frac{30x^4}{3x^{-2}}$

Working out:

Divide the numerical coefficients.
$\frac{30}{3} = 10$
Apply the division rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ .
$\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^6$
Combine the simplified numerical and variable terms.
$10x^6$
Answer:
$\boxed{10x^6}$
2. Simplify $\left(\frac{x^3}{2^2}\right)^{-2}$

Working out:
Apply the negative exponent rule, which states that $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .
$\left(\frac{x^3}{2^2}\right)^{-2} = \left(\frac{2^2}{x^3}\right)^2$
Apply the power of a quotient rule, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
$\left(\frac{2^2}{x^3}\right)^2 = \frac{(2^2)^2}{(x^3)^2}$
Apply the power of a power rule, which states that $(a^m)^n = a^{mn}$ .
$\frac{(2^2)^2}{(x^3)^2} = \frac{2^{2 \times 2}}{x^{3 \times 2}} = \frac{2^4}{x^6}$
Calculate the numerical value $2^4$ .
$2^4 = 2 \times 2 \times 2 \times 2 = 16$
Substitute the numerical value back into the expression.
$\frac{16}{x^6}$
Answer:
$\boxed{\frac{16}{x^6}}$
3. Simplify the powers: $\frac{x^6y^2z^{-3}}{x^{-1}z^{-5}}$

Working out:
Apply the division rule for exponents ( $\frac{a^m}{a^n} = a^{m-n}$ ) for each variable.
For $x$ :
$\frac{x^6}{x^{-1}} = x^{6 - (-1)} = x^{6+1} = x^7$
For $y$ : The $y^2$ term is only in the numerator, so it remains as $y^2$ .
For $z$ :
$\frac{z^{-3}}{z^{-5}} = z^{-3 - (-5)} = z^{-3+5} = z^2$
Combine the simplified terms.
$x^7y^2z^2$
Answer:
$\boxed{x^7y^2z^2}$

Simplify the following algebraic expressions: 1. (30x4)/(3x⁻²) 2. ((x3)/(22))⁻² 3. x6y2z-3x-1z⁻⁵