Certainly! Let's solve these expressions step-by-step.
1. Simplify the expression [tex]\(\frac{14 x^2}{4 x}\)[/tex]
Step-by-step solution:
1. Start with the given fraction:
[tex]\[
\frac{14 x^2}{4 x}
\][/tex]
2. Factor out common terms in the numerator and the denominator. Both the numerator and the denominator share a common factor of [tex]\(x\)[/tex]:
[tex]\[
\frac{14 x^2}{4 x} = \frac{14 x \cdot x}{4 \cdot x}
\][/tex]
3. Cancel the common [tex]\(x\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{14 x \cdot x}{4 \cdot x} = \frac{14 x}{4}
\][/tex]
4. Now, we simplify the numerical fraction [tex]\(\frac{14}{4}\)[/tex]:
[tex]\[
\frac{14}{4} = \frac{7}{2}
\][/tex]
5. So the simplified expression is:
[tex]\[
\frac{7 x}{2}
\][/tex]
Final Result for Problem 1:
[tex]\[
\frac{14 x^2}{4 x} = \frac{7 x}{2}
\][/tex]
2. Simplify the expression [tex]\(\frac{\sqrt{81 a^2}}{9 a^2}\)[/tex]
Step-by-step solution:
1. Start with the given fraction:
[tex]\[
\frac{\sqrt{81 a^2}}{9 a^2}
\][/tex]
2. Simplify the expression inside the square root:
[tex]\[
\sqrt{81 a^2} = \sqrt{81} \cdot \sqrt{a^2}
\][/tex]
3. Calculate the square root of each term:
[tex]\[
\sqrt{81} = 9 \quad \text{and} \quad \sqrt{a^2} = |a|
\][/tex]
4. Substitute these values back into the expression:
[tex]\[
\frac{9 |a|}{9 a^2}
\][/tex]
5. Cancel the common factor of [tex]\(9\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{9 |a|}{9 a^2} = \frac{|a|}{a^2}
\][/tex]
6. Simplify the fraction by dividing [tex]\( |a| \)[/tex] by [tex]\( a^2 \)[/tex]:
[tex]\[
\frac{|a|}{a^2} = \frac{|a|}{a \cdot a} = \frac{|a|}{a} \cdot \frac{1}{a} = \frac{|a|}{a^2}
\][/tex]
7. Recognize that [tex]\(|a| \)[/tex] over [tex]\(a\)[/tex] simplifies to 1 when [tex]\(a\)[/tex] is positive:
[tex]\[
\frac{|a|}{a \cdot a} = \frac{1}{a}
\][/tex]
So the simplified expression is:
[tex]\[
\frac{1}{a}
\][/tex]
Final Result for Problem 2:
[tex]\[
\frac{\sqrt{81 a^2}}{9 a^2} = \frac{1}{a}
\][/tex]
These step-by-step solutions lead us to the final simplified expressions for both problems:
[tex]\[
\frac{14 x^2}{4 x} = \frac{7 x}{2} \quad \text{and} \quad \frac{\sqrt{81 a^2}}{9 a^2} = \frac{1}{a}
\][/tex]
