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Chapter 2: Problem 37

Simplify. \(\left(-\frac{3 t^{4} u^{9}}{2 v^{7}}\right)^{4}\)

Short Answer

Expert verified
\(\frac{81t^{16}u^{36}}{16v^{28}}\)

Step by step solution

01

Apply the power rule

Raise every term inside the brackets to the power of 4. \(\left(-\frac{3 t^{4} u^{9}}{2 v^{7}}\right)^{4} = \left(-1\right)^4 \times \frac{3^4 t^{4^4} u^{9^4}}{2^4 v^{7^4}}\)
02

Simplify the powers

Simplify the powers of -1, 3, 2, t, u, and v. \(\left(-1\right)^4 = 1\) (since any negative number raised to an even power is positive) \(3^4 = 81\) \(2^4 = 16\) \(t^{4^4} = t^{16}\) \(u^{9^4} = u^{36}\) \(v^{7^4} = v^{28}\)
03

Combine the simplified terms

Place the simplified terms back into the expression. \(= 1 \times \frac{81t^{16}u^{36}}{16v^{28}}\) Since the coefficient of the expression is 1, we can rewrite it as: \(\frac{81t^{16}u^{36}}{16v^{28}}\) Now, the expression is fully simplified and the final answer is: \(\boxed{\frac{81t^{16}u^{36}}{16v^{28}}}\)

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