1) $$ 2 k^{3} $$ k) $$ (2 k)^{-3} $$ m) $$ (-2 x)^{-3} $$ i) $$ 4 x^{-1} y^{-2} $$ 0) $$ -5 x^{-3} y y^{-2} $$ P) $$ 2^{-2} x^{6} y^{-3} $$

Question

UpStudy AI · Accepted Answer

1) Simplify $$2 k^3$$:

- The expression is already simplified.
- Final answer: $$2 k^3$$.

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k) Simplify $$(2k)^{-3}$$:

- Use the rule $$(ab)^n = a^n\,b^n$$:
  $$
  (2k)^{-3} = 2^{-3}\, k^{-3}.
  $$
- Write $$2^{-3}$$ as a fraction:
  $$
  2^{-3} = \frac{1}{2^3} = \frac{1}{8}.
  $$
- Thus,
  $$
  (2k)^{-3} = \frac{1}{8} k^{-3}.
  $$
- To express with positive exponents, recall that $$k^{-3} = \frac{1}{k^3}$$:
  $$
  \frac{1}{8} k^{-3} = \frac{1}{8k^3}.
  $$
- Final answer: $$\frac{1}{8k^3}$$.

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m) Simplify $$(-2x)^{-3}$$:

- Apply the rule $$(ab)^n = a^n\,b^n$$:
  $$
  (-2x)^{-3} = (-2)^{-3}\, x^{-3}.
  $$
- Evaluate $$(-2)^{-3}$$:
  $$
  (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}.
  $$
- Thus,
  $$
  (-2x)^{-3} = -\frac{1}{8}\, x^{-3}.
  $$
- Write with positive exponent for $$x$$:
  $$
  -\frac{1}{8}\, x^{-3} = -\frac{1}{8x^3}.
  $$
- Final answer: $$-\frac{1}{8x^3}$$.

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i) Simplify $$4x^{-1}y^{-2}$$:

- Convert negative exponents:
  $$
  x^{-1} = \frac{1}{x} \quad \text{and} \quad y^{-2} = \frac{1}{y^2}.
  $$
- Thus,
  $$
  4x^{-1}y^{-2} = 4 \cdot \frac{1}{x} \cdot \frac{1}{y^2} = \frac{4}{x y^2}.
  $$
- Final answer: $$\frac{4}{xy^2}$$.

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0) Simplify $$-5x^{-3}yy^{-2}$$:

- First, combine the $$y$$ terms. Notice:
  $$
  y \cdot y^{-2} = y^{1-2} = y^{-1}.
  $$
- The expression becomes:
  $$
  -5x^{-3}y^{-1}.
  $$
- Write with positive exponents:
  $$
  x^{-3} = \frac{1}{x^3} \quad \text{and} \quad y^{-1} = \frac{1}{y},
  $$
  so
  $$
  -5x^{-3}y^{-1} = -\frac{5}{x^3y}.
  $$
- Final answer: $$-\frac{5}{x^3y}$$.

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P) Simplify $$2^{-2}x^{6}y^{-3}$$:

- Convert $$2^{-2}$$:
  $$
  2^{-2} = \frac{1}{2^2} = \frac{1}{4}.
  $$
- Also, rewrite $$y^{-3}$$ as:
  $$
  y^{-3} = \frac{1}{y^3}.
  $$
- So the expression becomes:
  $$
  2^{-2}x^{6}y^{-3} = \frac{1}{4} x^6 \frac{1}{y^3} = \frac{x^6}{4y^3}.
  $$
- Final answer: $$\frac{x^6}{4y^3}$$.
1) $$2 k^3$$ k) $$\frac{1}{8k^3}$$ m) $$-\frac{1}{8x^3}$$ i) $$\frac{4}{xy^2}$$ 0) $$-\frac{5}{x^3y}$$ P) $$\frac{x^6}{4y^3}$$

k)
m)
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