Write the following as simply as possible without using negative indices and as a single fraction…

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Write the following as simply as possible without using negative indices and as a single fraction: 3−2x⁻²(xy)(1)/(2) + 1/x

ScanSolve AI Tutor · Accepted Answer

QUESTION 4. Write the following as simply as possible without using negative indices and as a single fraction. c) 3⁻²x⁻²(xy)^(1/2) + 1/x Step 1: Convert terms with negative exponents to positive exponents. 3⁻² = (1)/(3²) = 1/9; x⁻² = (1)/(x²) The numerator of the first term becomes (1)/(9x²). Step 2: Apply the exponent rule (ab)ⁿ = aⁿ bⁿ to the denominator of the first term. (xy)^(1/2) = x^(1/2)y^(1/2) Step 3: Rewrite the first fraction by combining the results from Step 1 and Step 2. (1)/(9x²)x^(1/2)y^(1/2) = 19x² x^(1/2)y^(1/2); Step 4: Combine the x terms in the denominator of the first fraction using the rule a^m · aⁿ = a^(m+n). x² · x^(1/2) = x^(2 + 1/2) = x^(4/2 + 1/2) = x^(5/2) So the first fraction is 19x^(5/2)y^(1/2). Step 5: The expression is now 19x^(5/2)y^(1/2) + 1/x. To combine these into a single fraction, find the least common denominator (LCD). The LCD for 9x^(5/2)y^(1/2) and x is 9x^(5/2)y^(1/2). Step 6: Adjust the second term to have the LCD. To change the denominator x to 9x^(5/2)y^(1/2), we multiply the numerator and denominator by 9x^(3/2)y^(1/2). 1/x = 1 · (9x^(3/2)y^(1/2))x · (9x^(3/2)y^(1/2)) = 9x^(3/2)y^(1/2)9x^(1+3/2)y^(1/2) = 9x^(3/2)y^(1/2)9x^(5/2)y^(1/2) Step 7: Add the two fractions. 19x^(5/2)y^(1/2) + 9x^(3/2)y^(1/2)9x^(5/2)y^(1/2) = 1 + 9x^(3/2)y^(1/2)9x^(5/2)y^(1/2) The final answer is 1 + 9x^3/2y^(1/2)9x^(5/2)y^(1/2).

Write the following as simply as possible without using negative indices and as a single fraction: 3-2x⁻²(xy)(1)/(2) + (1)/(x)

Answer

Step-by-step solution

Part 4

Part c· 7 steps
$\frac{1 + 9x^{\frac{3}{2}}y^{\frac{1}{2}}}{9x^{\frac{5}{2}}y^{\frac{1}{2}}}$

Convert terms with negative exponents to positive exponents.

Apply the exponent rule $(ab)^n = a^n b^n$ to the denominator of the first term.

Rewrite the first fraction by combining the results from Step 1 and Step 2.

Combine the $x$ terms in the denominator of the first fraction using the rule $a^m \cdot a^n = a^{m+n}$ .

The expression is now $\frac{1}{9x^{\frac{5}{2}}y^{\frac{1}{2}}} + \frac{1}{x}$ .

Adjust the second term to have the LCD.

Add the two fractions.

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