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If 8^((5)/(6))-8^((1)/(2))=8^(m) for some value of m, what is the value of m ? Choose 1 answer: (A) (1)/(3) (B) (1)/(2) (c) 1 (D) (5)/(3)

If 856−812=8m 8^{\frac{5}{6}}-8^{\frac{1}{2}}=8^{m} for some value of m m , what is the value of m m ?\newlineChoose 11 answer:\newline(A) 13 \frac{1}{3} \newlineB 12 \frac{1}{2} \newline(C) 11\newline(D) 53 \frac{5}{3}

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Full solution

Q. If 856−812=8m 8^{\frac{5}{6}}-8^{\frac{1}{2}}=8^{m} for some value of m m , what is the value of m m ?\newlineChoose 11 answer:\newline(A) 13 \frac{1}{3} \newlineB 12 \frac{1}{2} \newline(C) 11\newline(D) 53 \frac{5}{3}
  1. Recognize Base Property: Recognize that the base of all terms is 88, which allows us to use the properties of exponents to combine the terms.
  2. Rewrite Subtraction as Division: Rewrite the subtraction of the exponents as the division of the same base with the exponents. 856−8128^{\frac{5}{6}} - 8^{\frac{1}{2}} can be written as 856×8−128^{\frac{5}{6}} \times 8^{-\frac{1}{2}} because subtracting exponents with the same base is equivalent to dividing them, which in turn is equivalent to multiplying by the reciprocal exponent.
  3. Add Exponents: Add the exponents since the bases are the same.\newline856+(−12)=856−36=8268^{\frac{5}{6} + \left(-\frac{1}{2}\right)} = 8^{\frac{5}{6} - \frac{3}{6}} = 8^{\frac{2}{6}}
  4. Simplify Exponent: Simplify the exponent 26\frac{2}{6} to its lowest terms.\newline26\frac{2}{6} simplifies to 13\frac{1}{3}.
  5. Write Simplified Exponent: Write the simplified exponent with the base. 826=8138^{\frac{2}{6}} = 8^{\frac{1}{3}}
  6. Conclude Value of m: Conclude that the value of m is the simplified exponent.\newlineTherefore, m=13m = \frac{1}{3}.

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