EXERCISE 5.5 1. Simplify the following and leave the answers in expon $$ \begin{array}{llll} ext { a) }\left(2^{6} ight)^{5} & ext { b) }\left(3^{4} ight)^{3} & ext { c) }\left(12^{2} ight)^{7} \ ext { e) }\left(a^{3} ight)^{9} & ext { f) }\left(a^{2} b^{4} ight)^{5} & ext { g) }\left(a^{2} b^{3} c^{4} ight)^{6}\end{array} $$ 2. Simplify as far as possible: $$ \begin{array}{llll} ext { a) } 5^{2}\left(a^{4} ight)^{3} & ext { b) }\left(4^{2} ight)^{3} \div\left(2^{4} ight)^{3} & ext { c) } \ ext { 3. Simplify: } & ext { b) } 5 a^{4}-\left(2 a^{2} ight)^{2} & ext { c) } \ \begin{array}{llll} ext { a) }\left(2 a^{2} b^{3} c ight)^{3} & ext { b) } 3^{-2}+3 \frac{1}{3}-\left(\frac{3}{10} ight)^{-1} & ext { c) } \ ext { 4. Simplify as far as possible: } \ ext { a) } 3^{-1}-4^{-1} & & \end{array}\end{array} \begin{array}{l} ext { 5. Simplify }\left(5 a^{3} ight)^{2}-2\left(3 a^{3} ight)^{2}\end{array} $$

Question

EXERCISE 5.5 1. Simplify the following and leave the answers in expon $$ \begin{array}{llll}	ext { a) }\left(2^{6}ight)^{5} & 	ext { b) }\left(3^{4}ight)^{3} & 	ext { c) }\left(12^{2}ight)^{7} \ 	ext { e) }\left(a^{3}ight)^{9} & 	ext { f) }\left(a^{2} b^{4}ight)^{5} & 	ext { g) }\left(a^{2} b^{3} c^{4}ight)^{6}\end{array} $$ 2. Simplify as far as possible: $$ \begin{array}{llll}	ext { a) } 5^{2}\left(a^{4}ight)^{3} & 	ext { b) }\left(4^{2}ight)^{3} \div\left(2^{4}ight)^{3} & 	ext { c) } \ 	ext { 3. Simplify: } & 	ext { b) } 5 a^{4}-\left(2 a^{2}ight)^{2} & 	ext { c) } \ \begin{array}{llll}	ext { a) }\left(2 a^{2} b^{3} cight)^{3} & 	ext { b) } 3^{-2}+3 \frac{1}{3}-\left(\frac{3}{10}ight)^{-1} & 	ext { c) } \ 	ext { 4. Simplify as far as possible: } \ 	ext { a) } 3^{-1}-4^{-1} & & \end{array}\end{array} \begin{array}{l}	ext { 5. Simplify }\left(5 a^{3}ight)^{2}-2\left(3 a^{3}ight)^{2}\end{array} $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\left(3^{4}\right)^{3}$$
- step1: Multiply the exponents:
  $$3^{4\times 3}$$
- step2: Multiply the numbers:
  $$3^{12}$$
Calculate or simplify the expression $$ (12^2)^7 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(12^{2}\right)^{7}$$
- step1: Multiply the exponents:
  $$12^{2\times 7}$$
- step2: Multiply the numbers:
  $$12^{14}$$
Calculate or simplify the expression $$ 5^2 * (a^4)^3 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$5^{2}\left(a^{4}\right)^{3}$$
- step1: Multiply the exponents:
  $$5^{2}a^{4\times 3}$$
- step2: Multiply the numbers:
  $$5^{2}a^{12}$$
- step3: Evaluate the power:
  $$25a^{12}$$
Calculate or simplify the expression $$ (a^3)^9 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(a^{3}\right)^{9}$$
- step1: Multiply the exponents:
  $$a^{3\times 9}$$
- step2: Multiply the numbers:
  $$a^{27}$$
Calculate or simplify the expression $$ (2^6)^5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(2^{6}\right)^{5}$$
- step1: Multiply the exponents:
  $$2^{6\times 5}$$
- step2: Multiply the numbers:
  $$2^{30}$$
Calculate or simplify the expression $$ (a^2 * b^3 * c^4)^6 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(a^{2}b^{3}c^{4}\right)^{6}$$
- step1: Use the properties of exponents:
  $$\left(a^{2}\right)^{6}\left(b^{3}\right)^{6}\left(c^{4}\right)^{6}$$
- step2: Evaluate the power:
  $$a^{12}b^{18}c^{24}$$
Calculate or simplify the expression $$ (2 * a^2 * b^3 * c)^3 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(2a^{2}b^{3}c\right)^{3}$$
- step1: Use the properties of exponents:
  $$2^{3}\left(a^{2}\right)^{3}\left(b^{3}\right)^{3}c^{3}$$
- step2: Evaluate the power:
  $$8a^{6}b^{9}c^{3}$$
Calculate or simplify the expression $$ (a^2 * b^4)^5 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(a^{2}b^{4}\right)^{5}$$
- step1: Use the properties of exponents:
  $$\left(a^{2}\right)^{5}\left(b^{4}\right)^{5}$$
- step2: Evaluate the power:
  $$a^{10}b^{20}$$
Calculate or simplify the expression $$ (5 * a^3)^2 - 2 * (3 * a^3)^2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(5a^{3}\right)^{2}-2\left(3a^{3}\right)^{2}$$
- step1: Multiply the terms:
  $$\left(5a^{3}\right)^{2}-18a^{6}$$
- step2: Rewrite the expression:
  $$25a^{6}-18a^{6}$$
- step3: Collect like terms:
  $$\left(25-18\right)a^{6}$$
- step4: Subtract the numbers:
  $$7a^{6}$$
Calculate or simplify the expression $$ 3^(-1) - 4^(-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3^{-1}-4^{-1}$$
- step1: Calculate:
  $$\frac{1}{3}-4^{-1}$$
- step2: Calculate:
  $$\frac{1}{3}-\frac{1}{4}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{4}{3\times 4}-\frac{3}{4\times 3}$$
- step4: Multiply the numbers:
  $$\frac{4}{12}-\frac{3}{4\times 3}$$
- step5: Multiply the numbers:
  $$\frac{4}{12}-\frac{3}{12}$$
- step6: Transform the expression:
  $$\frac{4-3}{12}$$
- step7: Subtract the numbers:
  $$\frac{1}{12}$$
Calculate or simplify the expression $$ (4^2)^3 / (2^4)^3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(4^{2}\right)^{3}}{\left(2^{4}\right)^{3}}$$
- step1: Multiply the exponents:
  $$\frac{\left(4^{2}\right)^{3}}{2^{4\times 3}}$$
- step2: Multiply the exponents:
  $$\frac{4^{2\times 3}}{2^{4\times 3}}$$
- step3: Multiply the numbers:
  $$\frac{4^{6}}{2^{4\times 3}}$$
- step4: Multiply the numbers:
  $$\frac{4^{6}}{2^{12}}$$
- step5: Rewrite the expression:
  $$\frac{2^{12}}{2^{12}}$$
- step6: Reduce the fraction:
  $$1$$
Calculate or simplify the expression $$ 3^(-2) + 3 * (1/3) - (3/10)^(-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3^{-2}+3\left(\frac{1}{3}\right)-\left(\frac{3}{10}\right)^{-1}$$
- step1: Evaluate:
  $$3^{-2}+3\times \frac{1}{3}-\left(\frac{3}{10}\right)^{-1}$$
- step2: Multiply the numbers:
  $$3^{-2}+1-\left(\frac{3}{10}\right)^{-1}$$
- step3: Evaluate the power:
  $$\frac{1}{9}+1-\left(\frac{3}{10}\right)^{-1}$$
- step4: Evaluate the power:
  $$\frac{1}{9}+1-\frac{10}{3}$$
- step5: Reduce fractions to a common denominator:
  $$\frac{1}{9}+\frac{9}{9}-\frac{10\times 3}{3\times 3}$$
- step6: Multiply the numbers:
  $$\frac{1}{9}+\frac{9}{9}-\frac{10\times 3}{9}$$
- step7: Transform the expression:
  $$\frac{1+9-10\times 3}{9}$$
- step8: Multiply the numbers:
  $$\frac{1+9-30}{9}$$
- step9: Calculate:
  $$\frac{-20}{9}$$
- step10: Rewrite the fraction:
  $$-\frac{20}{9}$$
Let's simplify the expressions step by step.

### 1. Simplify the following and leave the answers in exponent form:

**a)** $$(2^{6})^{5}$$

Using the power of a power property $$(a^m)^n = a^{m \cdot n}$$:
$$
(2^{6})^{5} = 2^{6 \cdot 5} = 2^{30}
$$

**b)** $$(3^{4})^{3}$$

Using the same property:
$$
(3^{4})^{3} = 3^{4 \cdot 3} = 3^{12}
$$

**c)** $$(12^{2})^{7}$$

Using the property:
$$
(12^{2})^{7} = 12^{2 \cdot 7} = 12^{14}
$$

**e)** $$(a^{3})^{9}$$

Using the property:
$$
(a^{3})^{9} = a^{3 \cdot 9} = a^{27}
$$

**f)** $$(a^{2} b^{4})^{5}$$

Using the property:
$$
(a^{2} b^{4})^{5} = a^{2 \cdot 5} b^{4 \cdot 5} = a^{10} b^{20}
$$

**g)** $$(a^{2} b^{3} c^{4})^{6}$$

Using the property:
$$
(a^{2} b^{3} c^{4})^{6} = a^{2 \cdot 6} b^{3 \cdot 6} c^{4 \cdot 6} = a^{12} b^{18} c^{24}
$$

### 2. Simplify as far as possible:

**a)** $$5^{2}(a^{4})^{3}$$

Using the property:
$$
5^{2}(a^{4})^{3} = 5^{2} a^{4 \cdot 3} = 25 a^{12}
$$

**b)** $$\left(4^{2}\right)^{3} \div \left(2^{4}\right)^{3}$$

First, simplify each part:
$$
(4^{2})^{3} = 4^{2 \cdot 3} = 4^{6} = (2^{2})^{6} = 2^{12}
$$
$$
(2^{4})^{3} = 2^{4 \cdot 3} = 2^{12}
$$
Now divide:
$$
\frac{2^{12}}{2^{12}} = 1
$$

**c)** $$(2 a^{2} b^{3} c)^{3}$$

Using the property:
$$
(2 a^{2} b^{3} c)^{3} = 2^{3} a^{2 \cdot 3} b^{3 \cdot 3} c^{3} = 8 a^{6} b^{9} c^{3}
$$

### 3. Simplify:

**a)** $$3^{-2} + 3 \cdot \frac{1}{3} - \left(\frac{3}{10}\right)^{-1}$$

Calculating each term:
$$
3^{-2} = \frac{1}{9}, \quad 3 \cdot \frac{1}{3} = 1, \quad \left(\frac{3}{10}\right)^{-1} = \frac{10}{3}
$$
Now combine:
$$
\frac{1}{9} + 1 - \frac{10}{3} = \frac{1}{9} + \frac{9}{9} - \frac{30}{9} = \frac{1 + 9 - 30}{9} = \frac{-20}{9}
$$

**b)** $$3^{-1} - 4^{-1}$$

Calculating each term:
$$
3^{-1} = \frac{1}{3}, \quad 4^{-1} = \frac{1}{4}
$$
Now combine:
$$
\frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12}
$$

### 4. Simplify as far as possible:

**5)** $$\left(5 a^{3}\right)^{2} - 2\left(3 a^{3}\right)^{2}$$

Calculating each term:
$$
(5 a^{3})^{2} = 25 a^{6}, \quad (3 a^{3})^{2} = 9 a^{6}
$$
Now combine:
$$
25 a^{6} - 2 \cdot 9 a^{6} = 25 a^{6} - 18 a^{6} = 7 a^{6}
$$

### Final Results:
1. a) $$2^{30}$$, b) $$3^{12}$$, c) $$12^{14}$$, e) $$a^{27}$$, f) $$a^{10}b^{20}$$, g) $$a^{12}b^{18}c^{24}$$
2. a) $$25 a^{12}$$, b) $$1$$, c) $$8 a^{6} b^{9} c^{3}$$
3. a) $$-\frac{20}{9}$$, b) $$\frac{1}{12}$$
4. 5) $$7 a^{6}$$
1. **a)** $$2^{30}$$, **b)** $$3^{12}$$, **c)** $$12^{14}$$, **e)** $$a^{27}$$, **f)** $$a^{10}b^{20}$$, **g)** $$a^{12}b^{18}c^{24}$$ 2. **a)** $$25 a^{12}$$, **b)** $$1$$, **c)** $$8 a^{6} b^{9} c^{3}$$ 3. **a)** $$-\frac{20}{9}$$, **b)** $$\frac{1}{12}$$ 4. **5)** $$7 a^{6}$$

Real Tutor Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions