$$ \frac{11 / w}{} $$ Solve for $$ x^{\prime} $$ (a) $$ x^{-2}=1 / 9 $$ (b) $$ 2 x^{3 / 5}=54 $$ (c) $$ 9^{2 x-3}=27 $$

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Certainly! Let's solve for $$ x $$ in each of the given equations step by step.

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### (a) $$ x^{-2} = \frac{1}{9} $$

**Step 1:** Rewrite the equation using positive exponents.

$$ x^{-2} = \frac{1}{9} $$
$$ \frac{1}{x^{2}} = \frac{1}{9} $$

**Step 2:** Since the denominators are equal, set the denominators equal to each other.

$$ x^{2} = 9 $$

**Step 3:** Solve for $$ x $$ by taking the square root of both sides.

$$ x = \pm \sqrt{9} $$
$$ x = \pm 3 $$

**Solution:** $$ x = 3 $$ or $$ x = -3 $$

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### (b) $$ 2 x^{\frac{3}{5}} = 54 $$

**Step 1:** Isolate $$ x^{\frac{3}{5}} $$.

$$ 2 x^{\frac{3}{5}} = 54 $$
$$ x^{\frac{3}{5}} = \frac{54}{2} $$
$$ x^{\frac{3}{5}} = 27 $$

**Step 2:** To solve for $$ x $$, raise both sides of the equation to the reciprocal of the exponent, which is $$ \frac{5}{3} $$.

$$ \left( x^{\frac{3}{5}} \right)^{\frac{5}{3}} = 27^{\frac{5}{3}} $$
$$ x = 27^{\frac{5}{3}} $$

**Step 3:** Simplify $$ 27^{\frac{5}{3}} $$.

First, recognize that $$ 27 = 3^3 $$.

$$ x = \left(3^3\right)^{\frac{5}{3}} $$
$$ x = 3^{3 \times \frac{5}{3}} $$
$$ x = 3^{5} $$
$$ x = 243 $$

**Solution:** $$ x = 243 $$

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### (c) $$ 9^{2x - 3} = 27 $$

**Step 1:** Express both sides of the equation with the same base.

Notice that $$ 9 = 3^2 $$ and $$ 27 = 3^3 $$.

$$ (3^2)^{2x - 3} = 3^3 $$

**Step 2:** Use the power of a power property $$(a^m)^n = a^{m \times n}$$.

$$ 3^{2(2x - 3)} = 3^3 $$
$$ 3^{4x - 6} = 3^3 $$

**Step 3:** Since the bases are the same, set the exponents equal to each other.

$$ 4x - 6 = 3 $$

**Step 4:** Solve for $$ x $$.

$$ 4x = 3 + 6 $$
$$ 4x = 9 $$
$$ x = \frac{9}{4} $$
$$ x = 2.25 $$

**Solution:** $$ x = \frac{9}{4} $$ or $$ x = 2.25 $$

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Feel free to ask if you have any more questions or need further clarification!
Solutions: (a) $$ x = 3 $$ or $$ x = -3 $$ (b) $$ x = 243 $$ (c) $$ x = \frac{9}{4} $$ or $$ x = 2.25 $$

\( \frac{11 / w}{} \) Solve for \( x^{\prime} \) (a) \( x^{-2}=1 / 9 \) (b) \( 2 x^{3 / 5}=54 \) (c) \( 9^{2 x-3}=27 \)

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