Questions: Use the laws of exponents to simplify. a^(8/3) * a^(5/6)

Use the laws of exponents to simplify. a^(8/3) * a^(5/6)
Transcript text: Use the laws of exponents to simplify. \[ a^{\frac{8}{3}} \cdot a^{\frac{5}{6}} \]
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Solution

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Solution Steps

Step 1: Apply the Product of Powers Property

When multiplying expressions with the same base, add the exponents: \[ a^{\frac{8}{3}} \cdot a^{\frac{5}{6}} = a^{\frac{8}{3} + \frac{5}{6}}. \]

Step 2: Find a Common Denominator for the Exponents

To add the exponents \(\frac{8}{3}\) and \(\frac{5}{6}\), find a common denominator. The least common denominator of 3 and 6 is 6: \[ \frac{8}{3} = \frac{16}{6}. \]

Step 3: Add the Exponents

Add the exponents using the common denominator: \[ \frac{16}{6} + \frac{5}{6} = \frac{21}{6}. \]

Step 4: Simplify the Exponent

Simplify \(\frac{21}{6}\): \[ \frac{21}{6} = \frac{7}{2}. \]

Step 5: Write the Final Simplified Expression

The simplified expression is: \[ a^{\frac{7}{2}}. \]

Final Answer

\(\boxed{a^{\frac{7}{2}}}\)

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