Question

Simplify each expression:

$ⓐ\frac{y^{43}}{y^{37}}ⓑ\frac{{10}^{15}}{{10}^7}ⓒ\frac{m^7}{m^{15}}ⓓ\frac{9^8}{9^{19}}$

Short answer

The simplified expression of$\frac{y^{43}}{y^{37}}=y^6$

The simplified expression of$\frac{{10}^{15}}{{10}^7}={10}^8$

The simplified expression of$\frac{m^7}{m^{15}}=\frac{1}{m^8}$

The simplified expression oflocalid="1661180727085" $\frac{9^8}{9^{19}}=\frac{1}{9^{11}}$

Step-by-step solution

Step 1. Given Information 

In the given question we have to simplify each expression:

$ⓐ\frac{y^{43}}{y^{37}}ⓑ\frac{{10}^{15}}{{10}^7}ⓒ\frac{m^7}{m^{15}}ⓓ\frac{9^8}{9^{19}}$

Part(a) Step 1. Simplifying the expression y43y37 

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=a^{m-n}$when$m>n$

Part(a) Step 2. The given expression is y43y37 

Since $43>37$, there are more factors oflocalid="1661180739523" $y$ in the numerator.
Use Quotient Property,

$\frac{y^{43}}{y^{37}}=y^{43-37}$

Simplify.

$\frac{y^{43}}{y^{37}}=y^6$

Part(b) Step 1. Simplifying the expression  1015107

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=a^{m-n}$when$m>n$

Part(b) Step 2. The given expression is 1015107

Since $15>7$, there are more factors oflocalid="1661180759192" $10$ in the numerator.
Use Quotient Property,

$\frac{{10}^{15}}{{10}^7}={10}^{15-7}$

Simplify.

$\frac{{10}^{15}}{{10}^7}={10}^8$

Part(c) Step 1. Simplifying the expression m7m15

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=\frac{1}{a^{m-n}}$when$m<n$

Part(c) Step 2. The given expression is m7m15

Since $7<15$, there are more factors oflocalid="1661180778591" $m$ in the denominator.
Use Quotient Property,

localid="1644691302135" $\frac{m^7}{m^{15}}=\frac{1}{m^{15-7}}$

Simplify.

$\frac{m^7}{m^{15}}=\frac{1}{m^8}$

Part(d) Step 1. Simplifying the expression 98919

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=\frac{1}{a^{m-n}}$when$m<n$

Part(d) Step 2. The given expression is 98919

Since $8<19$, there are more factors of localid="1661180803112" $9$in the denominator.

Use Quotient Property,

$\frac{9^8}{9^{19}}=\frac{1}{9^{19-8}}$

Simplify.

localid="1661180810949" $\frac{9^8}{9^{19}}=\frac{1}{9^{11}}$