## Math 7 Chapter 1 Lesson 5: Powers of a rational number

## 1. Summary of theory

### 1.1. Power of a rational number

The nth power of a rational number x, denoted \(x^n\), is the product of n factors x (n is a natural number greater than 1).

\({x^n} = \underbrace {xxx..x}_{n\,\,\,extra\,\,number}\) \(( x \in Q,n \in \mathbb{N},n>1)\).

\(x^n\) read as x to n or x to n or nth powers of x; x is called the base, n is called the exponent.

**– Convention: \(x^1=x\)** and \(x^0=x \ \ \ \ \ \ (x \ne 0)\).

– When writing the rational number x in the form \(\frac{a}{b} \ \ \ \ (a,b\in Z,b\ne 0)\), We have:

\({\left( {\frac{a}{b}} \right)^n} = \underbrace {\frac{a}{b}.\frac{a}{b}…\frac{a }{b}}_{n \ excess \ number} = \frac{{\overbrace {aa..a}^{n \ excess \ number}}}{{\underbrace {bb..b}_{n \ excess \ number }}} = \frac{{{a^n}}}{{{b^n}}}\).

So \({\left( {\frac{a}{b}} \right)^n} = \frac{{{a^n}}}{{b^n}}}\).

### 1.2. Product and quotient of two powers with the same base

When multiplying two powers with the same base, we keep the base and add the two exponents.

\({x^m}. {x^n} = {x^{m + n}}\).

When dividing two powers with the same base other than 0, we keep the base and subtract the exponent of the dividing power from the exponent of the dividing power.

\({x^m}:{x^n} = {x^{m – n}}\) with \(x \ne 0,\,m \ge n\).

### 1.3. Power of exponent

When calculating powers of a power, we keep the base and multiply the two exponents.

\({\left( {{x^m}} \right)^n} = {x^{mn}}\).

## 2. Illustrated exercise

**Question 1: **Calculate \({\left( {\frac{{ – 3}}{5}} \right)^2};{\left( { – \frac{1}{3}} \right)^3};{\ left( { – 0.2} \right)^3};{\left( {1,2} \right)^0}\).

**Solution guide**

\({\left( {\frac{{ – 3}}{5}} \right)^2} = \left( {\frac{{ – 3}}{5}} \right).\left( { { \frac{{ – 3}}{5}} \right) = \frac{{3.3}}{{5.5}} = \frac{9}{{25}}\).

\({\left( { – \frac{1}{3}} \right)^3} = \left( { – \frac{1}{3}} \right).\left( { – \frac{ 1}{3}} \right).\left( { – \frac{1}{3}} \right) = – \frac{{1.1.1}}{{3.3.3}} = – \frac{ 1}{9}\).

\({\left( { – 0.2} \right)^3} = \left( { – 0.2} \right).\left( { – 0.2} \right).\left( { – 0.2} \right) = – \left( {0.2} \right).\left( {0.2} \right).\left( {0.2} \right) = – 0.008\)

\({\left( {1,2} \right)^0}=1\)

**Verse 2: **Proof of equality \({(a + b)^2} = {a^2} + 2ab + {b^2}\).

Apply, calculate \(A = {(2{x^3} + 3{y^2})^2}\).

**Solution guide**

**Method 1: **We have \({(a + b)^2} = (a + b)(a + b)\)

Applying the distributive property of multiplying rational numbers to addition, we have:

\((a + b)(a + b) = a(a + b) + b(a + b) = {a^2} + ab + ba + {b^2} = {a^2} + 2ab + {b^2}\).

**Method 2: **Using common factorization and going from the right side, we have:

\({a^2} + 2ab + {b^2} = {a^2} + ab + ab + {b^2} = a(a + b) + b(a + b) = (a + b) )(a + b) = {(a + b)^2}\)

Apply: \(A = {(2{x^3} + 3{y^2})^2} = {(2{x^3})^2} + 2(2{x^3})(3{y ^2}) + {(3{y^2})^2}\)

\( \Rightarrow A = 4{x^6} + 12{x^3}{y^2} + 9{y^4}\).

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Find all natural numbers n such that: \(2.32 \ge {2^n} > 8\).

**Verse 2: **Find x, know:

a) \(x:{\left( { – \frac{1}{2}} \right)^3} = – \frac{1}{2}\).

b) \(x. {\left( {\frac{3}{5}} \right)^3} = {\left( {\frac{3}{5}} \right)^4}\).

c) \(x. {\left( {-\frac{3}{5}} \right)^3} = {\left( {\frac{3}{5}} \right)^6}\)

**Question 3: **Find a 5-digit number, which is the square of a natural number and is written with the digits 0; first; 2; 2; 2

### 3.2. Multiple choice exercises

**Question 1: **Product of \({3^4}{.3^6}\) equal:

A. \(3^2\).

B. \(9^2\).

C. \({3^{10}}\).

D. \(9^6\).

**Verse 2: **Choose the correct answer in the following sentences: \({a^n}:{a^2}\) equal

A. \({a^{n – 2}}\).

B. \({a^{n + 2}}\).

C. \({a^{2n}}\).

D. \({\left( {a:a} \right)^{n – 2}}\).

**Question 3: **Find the value of n known: \({4^n} + {4^{n + 1}} = 80\)

A. 1.

B. 2.

C. 3.

D. 4.

**Question 4: **For \({\left( {2{\rm{x}} + 1} \right)^3} = – 8\). What is the value of x:

A. \(-3\).

B. \(- \frac{3}{2}\).

C. \(\frac{4}{3}\).

D. None of the values of x satisfy.

**Question 5: **The natural numbers n satisfy \({3.3^2} \le {3^n} < {3^5}\) to be?

A. n = 3.

B. n = 4.

C. \(n \in \left\{ {3;4} \right\}\).

D. \(n \in \left\{ {3;4;5} \right\}\).

## 4. Conclusion

Through this lesson, you should achieve the following goals:

– The concept of powers of a rational number with natural exponents; how to multiply, divide two powers with the same base, how to calculate powers of powers.

– Proficiently apply the rules of multiplication, division of two powers with the same base, and powers of powers in solving problems.

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