simplify. your answer should contain only positive exponents.
1) $2m^{2} \cdot 2m^{3}$ 2) $m^{4} \cdot 2m^{-3}$

3) $4r^{-3} \cdot 2r^{2}$ 4) $4n^{4} \cdot 2n^{-3}$

5) $2k^{4} \cdot 4k$ 6) $2x^{3}y^{-3} \cdot 2x^{-1}y^{3}$

7) $2y^{2} \cdot 3x$ 8) $4v^{3} \cdot vu^{2}$

9) $4a^{3}b^{2} \cdot 3a^{-4}b^{-3}$ 10) $x^{2}y^{-4} \cdot x^{3}y^{2}$

11) $(x^{2})^{0}$ 12) $(2x^{2})^{-4}$

13) $(4r^{0})^{4}$ 14) $(4a^{3})^{2}$

15) $(3k^{4})^{4}$ 16) $(4xy)^{-1}$

Question

simplify. your answer should contain only positive exponents.
1) $2m^{2} \cdot 2m^{3}$						2) $m^{4} \cdot 2m^{-3}$

3) $4r^{-3} \cdot 2r^{2}$						4) $4n^{4} \cdot 2n^{-3}$

5) $2k^{4} \cdot 4k$						6) $2x^{3}y^{-3} \cdot 2x^{-1}y^{3}$

7) $2y^{2} \cdot 3x$						8) $4v^{3} \cdot vu^{2}$

9) $4a^{3}b^{2} \cdot 3a^{-4}b^{-3}$					10) $x^{2}y^{-4} \cdot x^{3}y^{2}$

11) $(x^{2})^{0}$						12) $(2x^{2})^{-4}$

13) $(4r^{0})^{4}$						14) $(4a^{3})^{2}$

15) $(3k^{4})^{4}$						16) $(4xy)^{-1}$

Accepted Answer

### Problem 1: $2m^2 \cdot 2m^3$
# Explanation:
## Step1: Multiply coefficients
Multiply the coefficients $2$ and $2$.
$2 \times 2 = 4$
## Step2: Multiply variables (exponent rule)
Use the rule $a^m \cdot a^n = a^{m + n}$ for $m^2 \cdot m^3$.
$m^2 \cdot m^3 = m^{2 + 3} = m^5$
## Step3: Combine results
Combine the coefficient and the variable.
$4 \times m^5 = 4m^5$
# Answer: $4m^5$

### Problem 2: $m^4 \cdot 2m^{-3}$
# Explanation:
## Step1: Multiply coefficients
Multiply the coefficients $1$ (for $m^4$) and $2$.
$1 \times 2 = 2$
## Step2: Multiply variables (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $m^4 \cdot m^{-3}$.
$m^4 \cdot m^{-3} = m^{4 + (-3)} = m^{1} = m$
## Step3: Combine results
Combine the coefficient and the variable.
$2 \times m = 2m$
# Answer: $2m$

### Problem 3: $4r^{-3} \cdot 2r^2$
# Explanation:
## Step1: Multiply coefficients
Multiply $4$ and $2$.
$4 \times 2 = 8$
## Step2: Multiply variables (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $r^{-3} \cdot r^2$.
$r^{-3} \cdot r^2 = r^{-3 + 2} = r^{-1}$
## Step3: Convert to positive exponent
Use $a^{-n} = \frac{1}{a^n}$, so $r^{-1} = \frac{1}{r}$. Then multiply by the coefficient.
$8 \times \frac{1}{r} = \frac{8}{r}$
# Answer: $\frac{8}{r}$

### Problem 4: $4n^4 \cdot 2n^{-3}$
# Explanation:
## Step1: Multiply coefficients
Multiply $4$ and $2$.
$4 \times 2 = 8$
## Step2: Multiply variables (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $n^4 \cdot n^{-3}$.
$n^4 \cdot n^{-3} = n^{4 + (-3)} = n^{1} = n$
## Step3: Combine results
Combine the coefficient and the variable.
$8 \times n = 8n$
# Answer: $8n$

### Problem 5: $2k^4 \cdot 4k$
# Explanation:
## Step1: Multiply coefficients
Multiply $2$ and $4$.
$2 \times 4 = 8$
## Step2: Multiply variables (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $k^4 \cdot k^1$ (since $k = k^1$).
$k^4 \cdot k^1 = k^{4 + 1} = k^5$
## Step3: Combine results
Combine the coefficient and the variable.
$8 \times k^5 = 8k^5$
# Answer: $8k^5$

### Problem 6: $2x^3y^{-3} \cdot 2x^{-1}y^3$
# Explanation:
## Step1: Multiply coefficients
Multiply $2$ and $2$.
$2 \times 2 = 4$
## Step2: Multiply $x$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $x^3 \cdot x^{-1}$.
$x^3 \cdot x^{-1} = x^{3 + (-1)} = x^2$
## Step3: Multiply $y$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $y^{-3} \cdot y^3$.
$y^{-3} \cdot y^3 = y^{-3 + 3} = y^0 = 1$ (since $a^0 = 1$)
## Step4: Combine results
Multiply the coefficient, $x$-term, and $y$-term.
$4 \times x^2 \times 1 = 4x^2$
# Answer: $4x^2$

### Problem 7: $2y^2 \cdot 3x$
# Explanation:
## Step1: Multiply coefficients
Multiply $2$ and $3$.
$2 \times 3 = 6$
## Step2: Multiply variables
Since $y^2$ and $x$ are different variables, we just combine them.
$y^2 \cdot x = xy^2$
## Step3: Combine results
Combine the coefficient and the variables.
$6 \times xy^2 = 6xy^2$
# Answer: $6xy^2$

### Problem 8: $4v^3 \cdot vu^2$
# Explanation:
## Step1: Multiply coefficients
Multiply $4$ and $1$ (for $v$).
$4 \times 1 = 4$
## Step2: Multiply $v$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $v^3 \cdot v^1$ (since $v = v^1$).
$v^3 \cdot v^1 = v^{3 + 1} = v^4$
## Step3: Combine with $u^2$
The $u^2$ term remains as is. Combine all parts.
$4 \times v^4 \times u^2 = 4u^2v^4$
# Answer: $4u^2v^4$

### Problem 9: $4a^3b^2 \cdot 3a^{-4}b^{-3}$
# Explanation:
## Step1: Multiply coefficients
Multiply $4$ and $3$.
$4 \times 3 = 12$
## Step2: Multiply $a$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $a^3 \cdot a^{-4}$.
$a^3 \cdot a^{-4} = a^{3 + (-4)} = a^{-1}$
## Step3: Multiply $b$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $b^2 \cdot b^{-3}$.
$b^2 \cdot b^{-3} = b^{2 + (-3)} = b^{-1}$
## Step4: Convert to positive exponents
Use $a^{-n} = \frac{1}{a^n}$ for $a^{-1}$ and $b^{-1}$.
$a^{-1} = \frac{1}{a}$, $b^{-1} = \frac{1}{b}$
## Step5: Combine results
Multiply the coefficient and the fractions.
$12 \times \frac{1}{a} \times \frac{1}{b} = \frac{12}{ab}$
# Answer: $\frac{12}{ab}$

### Problem 10: $x^2y^{-4} \cdot x^3y^2$
# Explanation:
## Step1: Multiply $x$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $x^2 \cdot x^3$.
$x^2 \cdot x^3 = x^{2 + 3} = x^5$
## Step2: Multiply $y$-terms (exponent rule)
Use $a^m \cdot a^n = a^{m + n}$ for $y^{-4} \cdot y^2$.
$y^{-4} \cdot y^2 = y^{-4 + 2} = y^{-2}$
## Step3: Convert $y$-term to positive exponent
Use $a^{-n} = \frac{1}{a^n}$ for $y^{-2}$.
$y^{-2} = \frac{1}{y^2}$
## Step4: Combine results
Multiply the $x$-term and the fraction for $y$.
$x^5 \times \frac{1}{y^2} = \frac{x^5}{y^2}$
# Answer: $\frac{x^5}{y^2}$

### Problem 11: $(x^2)^0$
# Explanation:
## Step1: Apply exponent rule
Use the rule $(a^m)^n = a^{m \times n}$, but also recall that any non - zero number (or variable) to the power of $0$ is $1$. For $(x^2)^0$, using $a^0 = 1$ (where $a=x^2$ and $x
eq0$), we get:
$(x^2)^0 = 1$ (assuming $x
eq0$; if $x = 0$, the expression is $0^0$ which is undefined, but in the context of simplifying with positive exponents, we assume $x
eq0$)
# Answer: $1$

### Problem 12: $(2x^2)^{-4}$
# Explanation:
## Step1: Apply exponent rule for products
Use $(ab)^n=a^n b^n$ for $(2x^2)^{-4}$.
$(2x^2)^{-4}=2^{-4}\times(x^2)^{-4}$
## Step2: Simplify $2^{-4}$
Use $a^{-n}=\frac{1}{a^n}$, so $2^{-4}=\frac{1}{2^4}=\frac{1}{16}$
## Step3: Simplify $(x^2)^{-4}$
Use $(a^m)^n=a^{m\times n}$, so $(x^2)^{-4}=x^{2\times(-4)} = x^{-8}$
## Step4: Convert $x^{-8}$ to positive exponent
Use $a^{-n}=\frac{1}{a^n}$, so $x^{-8}=\frac{1}{x^8}$
## Step5: Combine results
Multiply the two fractions.
$\frac{1}{16}\times\frac{1}{x^8}=\frac{1}{16x^8}$
# Answer: $\frac{1}{16x^8}$

### Problem 13: $(4r^0)^4$
# Explanation:
## Step1: Simplify $r^0$
Use $a^0 = 1$ (for $a = r$ and $r
eq0$), so $r^0 = 1$. Then the expression becomes $(4\times1)^4=(4)^4$
## Step2: Calculate $4^4$
$4^4=4\times4\times4\times4 = 256$
# Answer: $256$

### Problem 14: $(4a^3)^2$
# Explanation:
## Step1: Apply exponent rule for products
Use $(ab)^n=a^n b^n$ for $(4a^3)^2$.
$(4a^3)^2 = 4^2\times(a^3)^2$
## Step2: Simplify $4^2$
$4^2=16$
## Step3: Simplify $(a^3)^2$
Use $(a^m)^n=a^{m\times n}$, so $(a^3)^2=a^{3\times2}=a^6$
## Step4: Combine results
Multiply the two results.
$16\times a^6 = 16a^6$
# Answer: $16a^6$

### Problem 15: $(3k^4)^4$
# Explanation:
## Step1: Apply exponent rule for products
Use $(ab)^n=a^n b^n$ for $(3k^4)^4$.
$(3k^4)^4=3^4\times(k^4)^4$
## Step2: Simplify $3^4$
$3^4 = 3\times3\times3\times3=81$
## Step3: Simplify $(k^4)^4$
Use $(a^m)^n=a^{m\times n}$, so $(k^4)^4=k^{4\times4}=k^{16}$
## Step4: Combine results
Multiply the two results.
$81\times k^{16}=81k^{16}$
# Answer: $81k^{16}$

### Problem 16: $(4xy)^{-1}$
# Explanation:
## Step1: Apply exponent rule for products
Use $(ab)^n=a^n b^n$ for $(4xy)^{-1}$.
$(4xy)^{-1}=4^{-1}\times x^{-1}\times y^{-1}$
## Step2: Convert to positive exponents
Use $a^{-n}=\frac{1}{a^n}$ for each term.
$4^{-1}=\frac{1}{4}$, $x^{-1}=\frac{1}{x}$, $y^{-1}=\frac{1}{y}$
## Step3: Combine results
Multiply the fractions.
$\frac{1}{4}\times\frac{1}{x}\times\frac{1}{y}=\frac{1}{4xy}$
# Answer: $\frac{1}{4xy}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \(2m^2 \cdot 2m^3\)

Step1: Multiply coefficients

Step2: Multiply variables (exponent rule)

Step3: Combine results

Step1: Multiply coefficients

Step2: Multiply variables (exponent rule)

Step3: Combine results

Step1: Multiply coefficients

Step2: Multiply variables (exponent rule)

Step3: Convert to positive exponent

Answer:

Problem 2: \(m^4 \cdot 2m^{-3}\)