Question

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (27/64)-4/3

Accepted Answer

Recognize that the expression is a power raised to another power: $$\left( \frac{27}{64} \$$right)^{-\frac{4}$${3}}. $$Use the property of exponents: $$\left( a^m ight)^n = a$$^{m \cdot n}$$, which means you can rewrite the expression as $$\left( \frac{27}{64} \$$right)^{-\frac{4}$${3}}$$ without change, but this helps us think about simplifying the base first. Rewrite the base numbers as powers of their prime factors: 27$$ = $$3^3$$ and $$64 = 4^3$ = 2^6$, but since 64$$ = $$2^6 is $$not a perfect cube, let's use $$64 = 4^3$ is incorrect. Actually, 64$$ = $$2^6$$, but since the exponent is $$-\frac{4}{3}$$, it's easier to write $$64 = 4^3$ is incorrect. Instead, write 64$$ = $$4^3 is $$wrong; the correct prime factorization is $$64 = 2^6$. So, rewrite the fraction as $$\left( \frac{$$3^3$$}{$$2^6$$} \$$right)^{-\frac{4}$${3}}$$. Apply the exponent to both numerator and denominator separately: $$\left( $$3^3$$ \$$right)^{-\frac{4}$${3}}$$ and $$\left( $$2^6$$ \$$right)^{-\frac{4}$${3}}$$. Use the power of a power rule: $$\left( a^m ight)^n = $$a^{m \cdot n}$$, so the numerator becomes $$3^{3 	imes -\frac{4}$${3}}$$ and the denominator becomes 2$$^{6 	imes -\frac{4}$${3}}. $$Simplify the exponents by multiplying: For the numerator, $$3 	imes -\frac{4}{3} = -4$$, so the numerator is $$3^{-4}. $$For the denominator, $$6 	imes -\frac{4}{3} = -8$$, so the denominator is $$2^{-8}. $$The expression is now $$\frac{3$$^{-4}$$}{2$$^{-8}$$}. $$Rewrite the expression to eliminate negative exponents by using the rule $$a^{-m}$$ = \frac{1}{a^m}$$. So, 3$$^{-4}$$ = \frac{1}{3^4$}$$ and $$2^{-8}$$ = \frac{1}{$$2^8$$}$$. Since the denominator has a negative exponent, it moves to the numerator as a positive exponent. Therefore, the expression becomes $$\frac{1}{$$3^4$$} 	imes $$2^8$$, which can be written as $$\frac{2^8$}{3^4$}.$$

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (27/64)^-4/3

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Negative Exponents

Rational Exponents

Properties of Exponents and Radicals