7. Use the exponent rules to simplify the following products. Leave your answers in exponential form. a) $$ 3^{5} imes 3^{6} $$ b) $$ 5^{2} imes 5^{6} imes 5^{0} $$ c) $$ 11 imes 11^{4} $$ d) $$ 13^{n} imes 13^{2} $$ e) $$ y^{5} x y^{5} $$ f) $$ a^{2} imes a^{3} imes a^{+} $$ 2. Use the exponent rules to simplify the following quotients. Leave your answers in exponential form. a) $$ 8^{5} \div 8^{2} $$ b) $$ 7^{12} \div 7 $$ c) $$ b^{6} \div b^{3} $$ d) $$ \frac{9}{9} $$ e) $$ \frac{12^{\prime}}{12^{2}} $$ f) $$ \frac{1}{b^{1}} $$ 3. Use the exponent rules to simplify these, and leave your answers in exponential form. a) $$ \left(8^{6} ight)^{2} $$ b) $$ \left(2^{(2} ight)^{4} $$ c) $$ \left(a^{3} ight)^{3} $$ e) $$ 2\left(a^{10} ight)^{3} $$ d) $$ \left(p^{2} ight)^{2} $$ 4. Determine these values. f) $$ 3^{3}\left(\mathrm{y}^{2} ight)^{2} $$ a) $$ (-4)^{3}+\left(3^{2} ight)^{3} $$ b) $$ 3^{3} \div 3^{3}+\sqrt[3]{8} $$ c) $$ 4^{2}+4^{1}+4^{0} $$ e) $$ 10^{2} imes 10^{9} imes 10^{9}-10^{3} $$ d) $$ 3^{10}+3^{5} \div 3^{2} $$ f) $$ 3 a^{2} imes 2 a^{2} $$, 5. Simplify. b) $$ \left(3 x^{-2} ight)^{3} $$ a) $$ \left(x^{4} ight)^{5} $$ d) $$ \left(\frac{11}{1} ight)^{2} $$

Question

7. Use the exponent rules to simplify the following products. Leave your answers in exponential form. a) $$ 3^{5} 	imes 3^{6} $$ b) $$ 5^{2} 	imes 5^{6} 	imes 5^{0} $$ c) $$ 11 	imes 11^{4} $$ d) $$ 13^{n} 	imes 13^{2} $$ e) $$ y^{5} x y^{5} $$ f) $$ a^{2} 	imes a^{3} 	imes a^{+} $$ 2. Use the exponent rules to simplify the following quotients. Leave your answers in exponential form. a) $$ 8^{5} \div 8^{2} $$ b) $$ 7^{12} \div 7 $$ c) $$ b^{6} \div b^{3} $$ d) $$ \frac{9}{9} $$ e) $$ \frac{12^{\prime}}{12^{2}} $$ f) $$ \frac{1}{b^{1}} $$ 3. Use the exponent rules to simplify these, and leave your answers in exponential form. a) $$ \left(8^{6}ight)^{2} $$ b) $$ \left(2^{(2}ight)^{4} $$ c) $$ \left(a^{3}ight)^{3} $$ e) $$ 2\left(a^{10}ight)^{3} $$ d) $$ \left(p^{2}ight)^{2} $$ 4. Determine these values. f) $$ 3^{3}\left(\mathrm{y}^{2}ight)^{2} $$ a) $$ (-4)^{3}+\left(3^{2}ight)^{3} $$ b) $$ 3^{3} \div 3^{3}+\sqrt[3]{8} $$ c) $$ 4^{2}+4^{1}+4^{0} $$ e) $$ 10^{2} 	imes 10^{9} 	imes 10^{9}-10^{3} $$ d) $$ 3^{10}+3^{5} \div 3^{2} $$ f) $$ 3 a^{2} 	imes 2 a^{2} $$, 5. Simplify. b) $$ \left(3 x^{-2}ight)^{3} $$ a) $$ \left(x^{4}ight)^{5} $$ d) $$ \left(\frac{11}{1}ight)^{2} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\left(a^{3}ight)^{3}$$
- step1: Multiply the exponents:
  $$a^{3	imes 3}$$
- step2: Multiply the numbers:
  $$a^{9}$$
Calculate or simplify the expression $$ 8^{5} / 8^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{8^{5}}{8^{2}}$$
- step1: Multiply by $$a^{-n}:$$
  $$8^{5}	imes 8^{-2}$$
- step2: Multiply:
  $$8^{5-2}$$
- step3: Subtract the numbers:
  $$8^{3}$$
- step4: Evaluate the power:
  $$512$$
Calculate or simplify the expression $$ (11/1)^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{11}{1}ight)^{2}$$
- step1: Divide the terms:
  $$11^{2}$$
- step2: Evaluate the power:
  $$121$$
Calculate or simplify the expression $$ 4^{2} + 4^{1} + 4^{0} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$4^{2}+4^{1}+4^{0}$$
- step1: Evaluate the power:
  $$4^{2}+4^{1}+1$$
- step2: Evaluate the power:
  $$4^{2}+4+1$$
- step3: Evaluate the power:
  $$16+4+1$$
- step4: Add the numbers:
  $$21$$
Calculate or simplify the expression $$ 2 * (a^{10})^{3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2\left(a^{10}ight)^{3}$$
- step1: Multiply the exponents:
  $$2a^{10	imes 3}$$
- step2: Multiply the numbers:
  $$2a^{30}$$
Calculate or simplify the expression $$ (p^{2})^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(p^{2}ight)^{2}$$
- step1: Multiply the exponents:
  $$p^{2	imes 2}$$
- step2: Multiply the numbers:
  $$p^{4}$$
Calculate or simplify the expression $$ 3^{3} * (y^{2})^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$3^{3}\left(y^{2}ight)^{2}$$
- step1: Multiply the exponents:
  $$3^{3}y^{2	imes 2}$$
- step2: Multiply the numbers:
  $$3^{3}y^{4}$$
- step3: Evaluate the power:
  $$27y^{4}$$
Calculate or simplify the expression $$ (-4)^{3} + (3^{2})^{3} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-4ight)^{3}+\left(3^{2}ight)^{3}$$
- step1: Multiply the exponents:
  $$\left(-4ight)^{3}+3^{2	imes 3}$$
- step2: Multiply the numbers:
  $$\left(-4ight)^{3}+3^{6}$$
- step3: Simplify:
  $$-4^{3}+3^{6}$$
- step4: Evaluate the power:
  $$-64+3^{6}$$
- step5: Evaluate the power:
  $$-64+729$$
- step6: Add the numbers:
  $$665$$
Calculate or simplify the expression $$ 3 * a^{2} * 2 * a^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$3a^{2}	imes 2a^{2}$$
- step1: Multiply the terms:
  $$6a^{2}	imes a^{2}$$
- step2: Multiply the terms:
  $$6a^{2+2}$$
- step3: Add the numbers:
  $$6a^{4}$$
Calculate or simplify the expression $$ 10^{2} * 10^{9} * 10^{9} - 10^{3} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$10^{2}	imes 10^{9}	imes 10^{9}-10^{3}$$
- step1: Multiply:
  $$10^{20}-10^{3}$$
- step2: Evaluate the power:
  $$10^{20}-1000$$
Calculate or simplify the expression $$ 11 * 11^{4} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$11	imes 11^{4}$$
- step1: Calculate:
  $$11^{5}$$
Calculate or simplify the expression $$ 3^{10} + 3^{5} / 3^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3^{10}+\frac{3^{5}}{3^{2}}$$
- step1: Multiply by $$a^{-n}:$$
  $$3^{10}+3^{5}	imes 3^{-2}$$
- step2: Multiply the numbers:
  $$3^{10}+3^{3}$$
- step3: Evaluate the power:
  $$59049+3^{3}$$
- step4: Evaluate the power:
  $$59049+27$$
- step5: Add the numbers:
  $$59076$$
Calculate or simplify the expression $$ 3^{5} * 3^{6} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3^{5}	imes 3^{6}$$
- step1: Multiply:
  $$3^{5+6}$$
- step2: Add the numbers:
  $$3^{11}$$
Calculate or simplify the expression $$ 5^{2} * 5^{6} * 5^{0} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$5^{2}	imes 5^{6}	imes 5^{0}$$
- step1: Evaluate the power:
  $$5^{2}	imes 5^{6}	imes 1$$
- step2: Rewrite the expression:
  $$5^{2}	imes 5^{6}$$
- step3: Multiply:
  $$5^{2+6}$$
- step4: Add the numbers:
  $$5^{8}$$
Calculate or simplify the expression $$ 3^{3} / 3^{3} + 2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3^{3}}{3^{3}}+2$$
- step1: Divide the terms:
  $$1+2$$
- step2: Add the numbers:
  $$3$$
Calculate or simplify the expression $$ y^{5} * x * y^{5} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$y^{5}xy^{5}$$
- step1: Multiply the terms:
  $$y^{5+5}x$$
- step2: Add the numbers:
  $$y^{10}x$$
Calculate or simplify the expression $$ 13^{n} * 13^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$13^{n}	imes 13^{2}$$
- step1: Calculate:
  $$13^{n+2}$$
Calculate or simplify the expression $$ a^{2} * a^{3} * a^{+} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$a^{2}	imes a^{3}	imes a$$
- step1: Multiply the terms:
  $$a^{2+3+1}$$
- step2: Add the numbers:
  $$a^{6}$$
Calculate or simplify the expression $$ (3 * x^{-2})^{3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(3x^{-2}ight)^{3}$$
- step1: Use the properties of exponents:
  $$3^{3}\left(x^{-2}ight)^{3}$$
- step2: Evaluate the power:
  $$27x^{-6}$$
- step3: Simplify:
  $$\frac{27}{x^{6}}$$
Calculate or simplify the expression $$ 12^{1} / 12^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{12^{1}}{12^{2}}$$
- step1: Multiply by $$a^{-n}:$$
  $$12	imes 12^{-2}$$
- step2: Calculate:
  $$12^{-1}$$
- step3: Rewrite the expression:
  $$\frac{1}{12}$$
Calculate or simplify the expression $$ (x^{4})^{5} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(x^{4}ight)^{5}$$
- step1: Multiply the exponents:
  $$x^{4	imes 5}$$
- step2: Multiply the numbers:
  $$x^{20}$$
Calculate or simplify the expression $$ 7^{12} / 7 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{7^{12}}{7}$$
- step1: Divide the numbers:
  $$7^{12-1}$$
- step2: Subtract the terms:
  $$7^{11}$$
Calculate or simplify the expression $$ 1 / b^{1} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{1}{b}$$
Calculate or simplify the expression $$ b^{6} / b^{3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{b^{6}}{b^{3}}$$
- step1: Reduce the fraction:
  $$\frac{b^{6-3}}{1}$$
- step2: Simplify:
  $$b^{6-3}$$
- step3: Divide the terms:
  $$b^{3}$$
Calculate or simplify the expression $$ (2^{2})^{4} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(2^{2}ight)^{4}$$
- step1: Multiply the exponents:
  $$2^{2	imes 4}$$
- step2: Multiply the numbers:
  $$2^{8}$$
- step3: Evaluate the power:
  $$256$$
Calculate or simplify the expression $$ 9 / 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{9}{9}$$
- step1: Divide the terms:
  $$1$$
Calculate or simplify the expression $$ (8^{6})^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(8^{6}ight)^{2}$$
- step1: Multiply the exponents:
  $$8^{6	imes 2}$$
- step2: Multiply the numbers:
  $$8^{12}$$
Here are the simplified results for each of the problems you provided:

### 1. Simplifying Products
a) $$ 3^{5} 	imes 3^{6} = 3^{11} $$

b) $$ 5^{2} 	imes 5^{6} 	imes 5^{0} = 5^{8} $$

c) $$ 11 	imes 11^{4} = 11^{5} $$

d) $$ 13^{n} 	imes 13^{2} = 13^{n+2} $$

e) $$ y^{5} x y^{5} = y^{10} x $$

f) $$ a^{2} 	imes a^{3} 	imes a^{+} = a^{6} $$

### 2. Simplifying Quotients
a) $$ 8^{5} \div 8^{2} = 8^{3} $$

b) $$ 7^{12} \div 7 = 7^{11} $$

c) $$ b^{6} \div b^{3} = b^{3} $$

d) $$ \frac{9}{9} = 1 $$

e) $$ \frac{12^{1}}{12^{2}} = \frac{1}{12} $$

f) $$ \frac{1}{b^{1}} = \frac{1}{b} $$

### 3. Simplifying Powers
a) $$ (8^{6})^{2} = 8^{12} $$

b) $$ (2^{2})^{4} = 2^{8} $$

c) $$ (a^{3})^{3} = a^{9} $$

d) $$ 2 \left(a^{10}ight)^{3} = 2a^{30} $$

e) $$ (p^{2})^{2} = p^{4} $$

### 4. Determining Values
f) $$ 3^{3} \left(y^{2}ight)^{2} = 27y^{4} $$

a) $$ (-4)^{3} + (3^{2})^{3} = -64 + 27 = -37 $$

b) $$ 3^{3} \div 3^{3} + \sqrt[3]{8} = 1 + 2 = 3 $$

c) $$ 4^{2} + 4^{1} + 4^{0} = 16 + 4 + 1 = 21 $$

d) $$ 10^{2} 	imes 10^{9} 	imes 10^{9} - 10^{3} = 10^{20} - 10^{3} $$

e) $$ 3^{10} + 3^{5} \div 3^{2} = 59076 $$

f) $$ 3 a^{2} 	imes 2 a^{2} = 6 a^{4} $$

### 5. Simplifying Expressions
a) $$ (3 x^{-2})^{3} = \frac{27}{x^{6}} $$

b) $$ (x^{4})^{5} = x^{20} $$

c) $$ \left(\frac{11}{1}ight)^{2} = 121 $$

These results are left in exponential form where applicable. If you have any further questions or need additional assistance, feel free to ask!
Here are the simplified results for each problem: 1. **Simplifying Products** - a) $$ 3^{11} $$ - b) $$ 5^{8} $$ - c) $$ 11^{5} $$ - d) $$ 13^{n+2} $$ - e) $$ y^{10} x $$ - f) $$ a^{6} $$ 2. **Simplifying Quotients** - a) $$ 8^{3} $$ - b) $$ 7^{11} $$ - c) $$ b^{3} $$ - d) $$ 1 $$ - e) $$ \frac{1}{12} $$ - f) $$ \frac{1}{b} $$ 3. **Simplifying Powers** - a) $$ 8^{12} $$ - b) $$ 2^{8} $$ - c) $$ a^{9} $$ - d) $$ 2a^{30} $$ - e) $$ p^{4} $$ 4. **Determining Values** - f) $$ 27y^{4} $$ - a) $$ -37 $$ - b) $$ 3 $$ - c) $$ 21 $$ - d) $$ 10^{20} - 10^{3} $$ - e) $$ 59076 $$ - f) $$ 6a^{4} $$ 5. **Simplifying Expressions** - a) $$ \frac{27}{x^{6}} $$ - b) $$ x^{20} $$ - c) $$ 121 $$ These simplified answers are in exponential form where applicable.

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