Question

cl 3-118. simplify each expression.
a. $(5x^3)^2$
b. $\frac{14a^3b^2}{21a^4b}$
c. $2m^3n^2 \cdot 3mn^4$

Explanation:

Part a: Simplify \((5x^{3})^{2}\)

Step 1: Apply the power of a product rule

The power of a product rule states that \((ab)^n=a^n b^n\). So for \((5x^{3})^{2}\), we can apply this rule as follows:
\((5x^{3})^{2}=5^{2}\times(x^{3})^{2}\)

Step 2: Simplify each term

We know that \(5^{2} = 25\) and by the power of a power rule \((a^{m})^{n}=a^{mn}\), so \((x^{3})^{2}=x^{3\times2}=x^{6}\).
Combining these results, we get \(25x^{6}\).

Part b: Simplify \(\frac{14a^{3}b^{2}}{21a^{4}b}\)

Step 1: Simplify the coefficients

First, find the greatest common divisor (GCD) of 14 and 21. The GCD of 14 and 21 is 7. Divide both the numerator and the denominator by 7:
\(\frac{14\div7}{21\div7}=\frac{2}{3}\)

Step 2: Simplify the variables with the same base

For the variable \(a\), use the quotient rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\). Here, \(m = 3\) and \(n=4\), so \(\frac{a^{3}}{a^{4}}=a^{3-4}=a^{- 1}=\frac{1}{a}\) (we can also leave it as \(a^{-1}\), but usually we prefer positive exponents in the denominator).
For the variable \(b\), use the quotient rule of exponents \(\frac{b^{2}}{b^{1}}=b^{2 - 1}=b^{1}=b\)

Step 3: Combine the results

Multiply the simplified coefficient and the simplified variables: \(\frac{2}{3}\times\frac{1}{a}\times b=\frac{2b}{3a}\)

Part c: Simplify \(2m^{3}n^{2}\cdot3mn^{4}\)

Step 1: Multiply the coefficients

Multiply the coefficients 2 and 3: \(2\times3 = 6\)

Step 2: Multiply the variables with the same base

For the variable \(m\), use the product rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\). Here, \(m = 3\) and \(n = 1\) (since \(m=m^{1}\)), so \(m^{3}\cdot m^{1}=m^{3 + 1}=m^{4}\)
For the variable \(n\), use the product rule of exponents. Here, \(m = 2\) and \(n = 4\), so \(n^{2}\cdot n^{4}=n^{2+4}=n^{6}\)

Step 3: Combine the results

Multiply the simplified coefficient and the simplified variables: \(6\times m^{4}\times n^{6}=6m^{4}n^{6}\)

Answer:

a. \(\boldsymbol{25x^{6}}\)

b. \(\boldsymbol{\frac{2b}{3a}}\)

c. \(\boldsymbol{6m^{4}n^{6}}\)