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Chapter 2: Problem 16

Simplify. \(\left(-\frac{2}{5} z^{5}\right)^{3}(10 z)^{2}\)

Short Answer

Expert verified
The simplified expression is \(-\frac{32}{5}z^{17}\).

Step by step solution

01

Raise the fraction to the power of 3

To do this, we raise both the numerator and the denominator to the power of 3: \[\left(-\frac{2}{5} z^{5}\right)^{3} = -\frac{2^3}{5^3}z^{5 \cdot 3}\]
02

Compute the fraction

\[-\frac{2^3}{5^3} = -\frac{8}{125}\]
03

Multiply the exponent 5 by 3

\[z^{5\cdot 3}=z^{15}\]
04

Raise the factor (10z) to the power of 2

To do this, we raise both the coefficient (10) and the variable (z) to the power of 2: \[(10z)^2=10^2z^2\]
05

Compute 10 squared

\[10^2=100\]
06

Multiply the squared variable by z^2

\[z^2\]
07

Combine the results from the previous steps

Now, we can multiply the simplified forms from steps 2, 3, 5, and 6 together: \[-\frac{8}{125}z^{15}\cdot 100z^2\]
08

Simplify

Multiply the fractions and add the exponents of z: \[= -\frac{8 \cdot 100}{125}z^{15+2} = -\frac{800}{125}z^{17}\]
09

Further simplify the fraction

Since 800 and 125 have a common factor of 25, you could simplify the fraction further: \[-\frac{800}{125}=\frac{-800 \div 25}{125 \div 25} = -\frac{32}{5}\]
10

Final Answer

The simplified expression is: \[-\frac{32}{5}z^{17}\]

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