Question

Simplify each expression and eliminate any negative exponents.

1. $x^3\cdot x^4$
2. ${\left(2y^2\right)}^3$
3. localid="1651073187044" $y^{-2}{y}^7$
   

Short answer

1. $x^7$
2. $8y^6$

c. $y^5$

Step-by-step solution

a.Step-1 – Power of a number.

The power of a number represents how many times to use the number in a multiplication.

Step-2 – Laws of exponents.

If$a\ne 0$is a real number and n is a positive integer, then$a^0=1$and$a^{-n}=\frac{1}{a^n}$.

Multiplication rule:$a^m{a}^n=a^{m+n}$

Division rule:$\frac{a^m}{a^n}=a^{m-n}$

Power rule:${\left(a^x\right)}^y=a^{xy}$

Step-3 – Use the exponent rule into the given expression.

Use the multiplication rule of exponents:

$\begin{array}{l}{\mathbf{x}}^{\mathbf{3}}\mathbf{\cdot }{\mathbf{x}}^{\mathbf{4}}\mathbf{=}{\mathbf{x}}^{\mathbf{3}\mathbf{+}\mathbf{4}}\\ \mathbf{=}{\mathbf{x}}^{\mathbf{7}}\end{array}$

b.Step-1 – Power of a number.

The power of a number represents how many times to use the number in a multiplication.

Step-2 – Laws of exponents.

If$a\ne 0$is a real number and n is a positive integer, then$a^0=1$and$a^{-n}=\frac{1}{a^n}$.

Multiplication rule:$a^m{a}^n=a^{m+n}$

Division rule:$\frac{a^m}{a^n}=a^{m-n}$

Power rule:${\left(a^x\right)}^y=a^{xy}$

Step-3 – Use the exponent rule into the given expression.

Use the product rule of exponents:

$\begin{array}{l}{\left(2y^2\right)}^3=2^3\cdot y^{2\cdot 3}\\ =8y^6\end{array}$

c. Step-1 – Power of a number.

The power of a number represents how many times to use the number in a multiplication.

Step-2 – Laws of exponents.

If$a\ne 0$is a real number and n is a positive integer, then $a^0=1$and$a^{-n}=\frac{1}{a^n}$.

Multiplication rule:$a^m{a}^n=a^{m+n}$

Division rule:$\frac{a^m}{a^n}=a^{m-n}$

Power rule:${\left(a^x\right)}^y=a^{xy}$

Step-3 – Use the exponent rule into the given expression.

Use the Multiplication rule of exponents:

$\begin{array}{l}{y}^{-2}{y}^7=y^{-2+7}\\ =y^5\end{array}$