Question

simplify each exponential expression!

1. (3x^4y^5)(7x^5y) =
2. \frac{40a^6b^3}{8a^1b^8} =
3. (9m^3n)^2 =
4. circle all the rational numbers!!

a. 4/3
b. \sqrt{21}
c. 7\pi
d. 6.\overline{7}
e. \sqrt{144}

Explanation:

Step1: Multiply coefficients and add exponents of like - variables

For $(3x^{4}y^{5})(7x^{5}y)$, multiply the coefficients $3\times7 = 21$, and for the $x$ - terms $x^{4}\times x^{5}=x^{4 + 5}=x^{9}$, for the $y$ - terms $y^{5}\times y=y^{5+1}=y^{6}$. So, $(3x^{4}y^{5})(7x^{5}y)=21x^{9}y^{6}$.

Step2: Simplify the fraction using exponent rules

For $\frac{40a^{6}b^{3}}{8a^{1}b^{8}}$, first simplify the coefficient $\frac{40}{8}=5$. Then, for the $a$ - terms, $a^{6}\div a^{1}=a^{6 - 1}=a^{5}$, and for the $b$ - terms, $b^{3}\div b^{8}=b^{3-8}=b^{- 5}=\frac{1}{b^{5}}$. So, $\frac{40a^{6}b^{3}}{8a^{1}b^{8}}=\frac{5a^{5}}{b^{5}}$.

Step3: Expand the power of a product

For $(9m^{3}n)^{2}$, use the rule $(ab)^{n}=a^{n}b^{n}$. So, $(9m^{3}n)^{2}=9^{2}(m^{3})^{2}n^{2}=81m^{6}n^{2}$.

Step4: Recall the definition of rational numbers

A rational number is a number that can be written as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$.

• $\frac{4}{3}$ is a rational number.
• $\sqrt{21}$ is an irrational number since 21 is not a perfect - square.
• $7\pi$ is irrational because $\pi$ is irrational.
• $6.\overline{7}$ is a rational number as it is a repeating decimal and can be written as a fraction.
• $\sqrt{144}=12$, which is a rational number.

Answer:

1. $21x^{9}y^{6}$
2. $\frac{5a^{5}}{b^{5}}$
3. $81m^{6}n^{2}$
4. A. $\frac{4}{3}$, D. $6.\overline{7}$, E. $\sqrt{144}$