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Express the following fraction in simplest form, only using positive exponents. (3(b^(-4))^(3))/(15b^(-7)j^(-5)) Answer:

Express the following fraction in simplest form, only using positive exponents.\newline3(b−4)315b−7j−5 \frac{3\left(b^{-4}\right)^{3}}{15 b^{-7} j^{-5}} \newlineAnswer:

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Full solution

Q. Express the following fraction in simplest form, only using positive exponents.\newline3(b−4)315b−7j−5 \frac{3\left(b^{-4}\right)^{3}}{15 b^{-7} j^{-5}} \newlineAnswer:
  1. Write Original Expression: Write down the original expression.\newlineWe have the expression (3(b−4)315b−7j−5)(\frac{3(b^{-4})^3}{15b^{-7}j^{-5}}).
  2. Simplify Numerator: Simplify the numerator.\newlineThe numerator is 3(b−4)33(b^{-4})^3. According to the power of a power rule, we multiply the exponents.\newline3×(b−4×3)=3×b−12.3 \times (b^{-4 \times 3}) = 3 \times b^{-12}.
  3. Simplify Denominator: Simplify the denominator.\newlineThe denominator is 15b−7j−515b^{-7}j^{-5}. We can rewrite negative exponents as positive by taking the reciprocal of the base.\newline(15/(b7))(1/(j5))=15/(b7j5)(15/(b^7))(1/(j^5)) = 15/(b^7j^5).
  4. Combine Numerator and Denominator: Combine the numerator and denominator.\newlineNow we have (3⋅b−12)/(15b7j5)(3 \cdot b^{-12}) / (\frac{15}{b^7j^5}).\newlineWe can simplify this by multiplying the numerator by the reciprocal of the denominator.\newline(3⋅b−12)⋅(b7j515)(3 \cdot b^{-12}) \cdot (\frac{b^7j^5}{15}).
  5. Combine Like Terms: Simplify the expression by combining like terms. We can combine the bb terms by adding the exponents and simplify the constants by division. (315)⋅b(−12+7)⋅j5=(15)⋅b−5⋅j5(\frac{3}{15}) \cdot b^{(-12 + 7)} \cdot j^5 = (\frac{1}{5}) \cdot b^{-5} \cdot j^5.
  6. Rewrite with Positive Exponents: Rewrite the expression with positive exponents.\newlineTo express b−5b^{-5} with a positive exponent, we take the reciprocal of b5b^5.\newline(15)⋅(1b5)⋅j5=j55b5.(\frac{1}{5}) \cdot (\frac{1}{b^5}) \cdot j^5 = \frac{j^5}{5b^5}.

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