$$ \begin{array}{ll} ext { c) }\left(x^{2} y z^{-1} ight)^{3} imes\left(x^{4} y z^{3} ight)^{-1} & ext { d) }\left(\frac{1}{x y^{2}} ight)^{-2} imes\left(\frac{x^{3}}{y^{2}} ight)^{-1} \ \begin{array}{ll}\frac{a^{2} b^{-2} c^{3}}{a b^{-1} c^{2}} imes \frac{\left(a^{2} b^{-2} c ight)^{-1}}{a^{-1} b^{2} c^{3}} \div \frac{a b}{c^{4}} & ext { f) } \frac{f g}{h^{3}} imes\left(\frac{h^{4}}{f^{2} g} ight)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}} \ ext { e) }\left(\frac{1}{x}+\frac{x}{y} ight)^{-2} & ext { h) } \frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}} \ ext { g) } \frac{8^{2 x+2}+4^{3 x-1}}{2^{3 x}} & ext { j) } \frac{54^{x+1} 36^{x-1}}{24^{2 x-3}}\end{array}\end{array} $$

Question

$$ \begin{array}{ll}	ext { c) }\left(x^{2} y z^{-1}ight)^{3} 	imes\left(x^{4} y z^{3}ight)^{-1} & 	ext { d) }\left(\frac{1}{x y^{2}}ight)^{-2} 	imes\left(\frac{x^{3}}{y^{2}}ight)^{-1} \ \begin{array}{ll}\frac{a^{2} b^{-2} c^{3}}{a b^{-1} c^{2}} 	imes \frac{\left(a^{2} b^{-2} cight)^{-1}}{a^{-1} b^{2} c^{3}} \div \frac{a b}{c^{4}} & 	ext { f) } \frac{f g}{h^{3}} 	imes\left(\frac{h^{4}}{f^{2} g}ight)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}} \ 	ext { e) }\left(\frac{1}{x}+\frac{x}{y}ight)^{-2} & 	ext { h) } \frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}} \ 	ext { g) } \frac{8^{2 x+2}+4^{3 x-1}}{2^{3 x}} & 	ext { j) } \frac{54^{x+1} 36^{x-1}}{24^{2 x-3}}\end{array}\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's simplify each of the given expressions step by step.

---

### **c) $$\left(x^{2} y z^{-1}\right)^{3} \times \left(x^{4} y z^{3}\right)^{-1}$$**

**Step 1:** Apply the exponent to each factor inside the parentheses.

$$
\left(x^{2}\right)^{3} \times \left(y\right)^{3} \times \left(z^{-1}\right)^{3} \times \left(x^{4}\right)^{-1} \times \left(y\right)^{-1} \times \left(z^{3}\right)^{-1}
$$

**Step 2:** Simplify the exponents by multiplying them.

$$
x^{6} \times y^{3} \times z^{-3} \times x^{-4} \times y^{-1} \times z^{-3}
$$

**Step 3:** Combine like terms by adding the exponents.

$$
x^{6-4} \times y^{3-1} \times z^{-3-3} = x^{2} \times y^{2} \times z^{-6}
$$

**Step 4:** Rewrite with positive exponents.

$$
\frac{x^{2} y^{2}}{z^{6}}
$$

---

### **d) $$\left(\frac{1}{x y^{2}}\right)^{-2} \times \left(\frac{x^{3}}{y^{2}}\right)^{-1}$$**

**Step 1:** Apply the negative exponents by taking reciprocals and changing the sign of the exponents.

$$
(x y^{2})^{2} \times \left(\frac{y^{2}}{x^{3}}\right)
$$

**Step 2:** Expand the expressions.

$$
x^{2} y^{4} \times \frac{y^{2}}{x^{3}} 
$$

**Step 3:** Combine like terms by adding the exponents.

$$
\frac{x^{2}}{x^{3}} \times y^{4+2} = x^{-1} \times y^{6}
$$

**Step 4:** Rewrite with positive exponents.

$$
\frac{y^{6}}{x}
$$

---

### **e) $$\left(\frac{1}{x} + \frac{x}{y}\right)^{-2}$$**

**Step 1:** Combine the terms inside the parentheses by finding a common denominator.

$$
\frac{1 \cdot y + x \cdot x}{x y} = \frac{y + x^{2}}{x y}
$$

**Step 2:** Apply the exponent.

$$
\left(\frac{y + x^{2}}{x y}\right)^{-2} = \left(\frac{x y}{y + x^{2}}\right)^{2}
$$

**Step 3:** Square both the numerator and the denominator.

$$
\frac{(x y)^{2}}{(y + x^{2})^{2}} = \frac{x^{2} y^{2}}{(y + x^{2})^{2}}
$$

---

### **f) $$\frac{a^{2} b^{-2} c^{3}}{a b^{-1} c^{2}} \times \frac{\left(a^{2} b^{-2} c\right)^{-1}}{a^{-1} b^{2} c^{3}} \div \frac{a b}{c^{4}}$$**

**Step 1:** Simplify each fraction separately.

For the first fraction:

$$
\frac{a^{2}}{a} \times \frac{b^{-2}}{b^{-1}} \times \frac{c^{3}}{c^{2}} = a^{1} b^{-1} c^{1}
$$

For the second fraction:

$$
\frac{\left(a^{2} b^{-2} c\right)^{-1}}{a^{-1} b^{2} c^{3}} = \frac{a^{-2} b^{2} c^{-1}}{a^{-1} b^{2} c^{3}} = a^{-2 +1} b^{2-2} c^{-1-3} = a^{-1} c^{-4}
$$

**Step 2:** Combine the simplified fractions and perform the division.

$$
(a^{1} b^{-1} c^{1}) \times (a^{-1} c^{-4}) \div \frac{a b}{c^{4}} = (a^{1-1} b^{-1} c^{1-4}) \div \frac{a b}{c^{4}} = b^{-1} c^{-3} \div \frac{a b}{c^{4}}
$$

**Step 3:** Change division to multiplication by the reciprocal.

$$
b^{-1} c^{-3} \times \frac{c^{4}}{a b} = \frac{c^{4}}{a b^{2} c^{3}} = \frac{c^{1}}{a b^{2}} = \frac{c}{a b^{2}}
$$

---

### **g) $$\frac{8^{2x+2} + 4^{3x-1}}{2^{3x}}$$**

**Step 1:** Express all terms with base 2.

$$
8 = 2^{3}, \quad 4 = 2^{2}
$$

So,

$$
8^{2x+2} = (2^{3})^{2x+2} = 2^{6x +6}, \quad 4^{3x-1} = (2^{2})^{3x-1} = 2^{6x -2}
$$

**Step 2:** Substitute back into the expression.

$$
\frac{2^{6x +6} + 2^{6x -2}}{2^{3x}}
$$

**Step 3:** Factor out $$2^{6x -2}$$ from the numerator.

$$
\frac{2^{6x -2} \left(2^{8} + 1\right)}{2^{3x}} = 2^{6x -2 -3x} \times (256 + 1) = 2^{3x -2} \times 257
$$

$$
= 257 \times 2^{3x -2}
$$

---

### **h) $$\frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}}$$**

**Step 1:** Express all terms with prime bases.

$$
27 = 3^{3}, \quad 18 = 2 \times 3^{2}, \quad 36 = 2^{2} \times 3^{2}
$$

So,

$$
27^{x+1} = (3^{3})^{x+1} = 3^{3x +3}, \quad 18^{x-1} = (2 \times 3^{2})^{x-1} = 2^{x-1} \times 3^{2x -2}, \quad 36^{3-x} = (2^{2} \times 3^{2})^{3-x} = 2^{6 -2x} \times 3^{6 -2x}
$$

**Step 2:** Substitute back into the expression.

$$
\frac{3^{3x +3} \times 2^{x-1} \times 3^{2x -2}}{2^{6 -2x} \times 3^{6 -2x}} = \frac{2^{x-1} \times 3^{5x +1}}{2^{6 -2x} \times 3^{6 -2x}}
$$

**Step 3:** Simplify by subtracting exponents.

$$
2^{x-1 - (6 -2x)} \times 3^{5x +1 - (6 -2x)} = 2^{3x -7} \times 3^{7x -5}
$$

---

### **j) $$\frac{54^{x+1} \cdot 36^{x-1}}{24^{2x-3}}$$**

**Step 1:** Express all terms with prime bases.

$$
54 = 2 \times 3^{3}, \quad 36 = 2^{2} \times 3^{2}, \quad 24 = 2^{3} \times 3
$$

So,

$$
54^{x+1} = (2 \times 3^{3})^{x+1} = 2^{x+1} \times 3^{3(x+1)} = 2^{x+1} \times 3^{3x+3}
$$

$$
36^{x-1} = (2^{2} \times 3^{2})^{x-1} = 2^{2(x-1)} \times 3^{2(x-1)} = 2^{2x-2} \times 3^{2x-2}
$$

$$
24^{2x-3} = (2^{3} \times 3)^{2x-3} = 2^{3(2x-3)} \times 3^{2x-3} = 2^{6x-9} \times 3^{2x-3}
$$

**Step 2:** Substitute back into the expression.

$$
\frac{2^{x+1} \times 3^{3x+3} \times 2^{2x-2} \times 3^{2x-2}}{2^{6x-9} \times 3^{2x-3}} = \frac{2^{3x -1} \times 3^{5x +1}}{2^{6x -9} \times 3^{2x -3}}
$$

**Step 3:** Simplify by subtracting exponents.

$$
2^{3x -1 - (6x -9)} \times 3^{5x +1 - (2x -3)} = 2^{-3x +8} \times 3^{3x +4}
$$

$$
= 3^{3x +4} \times 2^{-3x +8}
$$

---

### **f) $$\frac{f g}{h^{3}} \times \left(\frac{h^{4}}{f^{2} g}\right)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}}$$**

**Step 1:** Simplify each part.

$$
\left(\frac{h^{4}}{f^{2} g}\right)^{-2} = \left(\frac{f^{2} g}{h^{4}}\right)^{2} = \frac{f^{4} g^{2}}{h^{8}}
$$

$$
\frac{a^{0} f^{2} g^{-2}}{h^{-3}} = a^{0} f^{2} g^{-2} \times h^{3} = f^{2} h^{3} g^{-2}
$$

**Step 2:** Substitute back into the expression.

$$
\frac{f g}{h^{3}} \times \frac{f^{4} g^{2}}{h^{8}} \div (f^{2} h^{3} g^{-2}) = \frac{f g \times f^{4} g^{2}}{h^{3} \times h^{8}} \div f^{2} h^{3} g^{-2}
$$

**Step 3:** Simplify multiplication.

$$
\frac{f^{5} g^{3}}{h^{11}} \div f^{2} h^{3} g^{-2}
$$

**Step 4:** Change division to multiplication by the reciprocal.

$$
\frac{f^{5} g^{3}}{h^{11}} \times \frac{1}{f^{2} h^{3} g^{-2}} = \frac{f^{5}}{f^{2}} \times \frac{g^{3}}{g^{-2}} \times \frac{1}{h^{11} h^{3}} = f^{3} g^{5} h^{-14}
$$

**Step 5:** Rewrite with positive exponents.

$$
\frac{f^{3} g^{5}}{h^{14}}
$$

---
Here are the simplified answers for each expression: **c)** $$ \frac{x^{2} y^{2}}{z^{6}} $$ **d)** $$ \frac{y^{6}}{x} $$ **e)** $$ \frac{c}{a b^{2}} $$ **f)** $$ 3^{3x +4} \times 2^{-3x +8} $$ **g)** $$ 257 \times 2^{3x -2} $$ **h)** $$ 2^{3x -7} \times 3^{7x -5} $$ **j)** $$ \frac{f^{3} g^{5}}{h^{14}} $$

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