Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Simplify expressions with rational exponents.
• Convert between radical and exponential forms.
• Solve simple exponential equations.
• Apply exponent laws to complex problems.

Lesson summary

This week, we're diving deeper into exponents, building on what you learned in previous grades. Exponents are a fundamental concept in mathematics and are used extensively in various fields, from finance and science to computer programming and even everyday calculations. Understanding exponents allows us to efficiently represent and manipulate very large or very small numbers. In South Africa, exponents are crucial in understanding concepts such as compound interest on loans and investments, population growth, and scientific data related to climate change and resource management.

Lesson notes

2.1 Rational Exponents A rational exponent is an exponent that is a fraction. For example, `x^(1/2)` or `y^(3/4)`. A rational exponent represents both a root and a power. The denominator of the fraction indicates the index of the root, and the numerator indicates the power to which the base is raised.

In general: `x^(m/n) = (n√x)^m = n√(x^m)` where n is the root and m is the power. `x^(1/n) = n√x` (This is a special case where m = 1). Important

Note: When dealing with even roots (square root, fourth root, etc.), remember that the base must be non-negative (greater than or equal to zero) to have a real number solution.

Example 1: Simplify `8^(2/3)` Explanation: We have a rational exponent of 2/

3. The denominator (3) tells us to take the cube root, and the numerator (2) tells us to raise the result to the power of

2. Solution: `8^(2/3) = (³√8)² = (2)² = 4` Example 2: Simplify `16^(3/4)` Explanation: We have a rational exponent of 3/

4. The denominator (4) tells us to take the fourth root, and the numerator (3) tells us to raise the result to the power of

3. Solution: `16^(3/4) = (⁴√16)³ = (2)³ = 8` 2.2 Converting Between Radical and Exponential Form It is crucial to be able to convert seamlessly between radical and exponential forms. This skill is vital for simplifying expressions and solving equations.

Radical to Exponential: If you have `n√x^m`, this is equivalent to `x^(m/n)`.

Exponential to Radical: If you have `x^(m/n)`, this is equivalent to `n√x^m` or `(n√x)^m`.

Example 1: Convert `√x³` to exponential form.

Explanation: Remember that a square root is the same as taking the root with index 2 (i.e., `√x = ²√x`). So, we have `²√x³`.

Solution: `√x³ = x^(3/2)` Example 2: Convert `5√(y²) `to exponential form.

Explanation: We have the fifth root of y squared.

Solution: `5√(y²) = y^(2/5)` Example 3: Convert `a^(5/7)` to radical form.

Explanation: The denominator (7) is the index of the root, and the numerator (5) is the power.

Solution: `a^(5/7) = ⁷√(a⁵)` 2.3 Solving Simple Exponential Equations An exponential equation is an equation where the variable appears in the exponent. To solve these equations (in simple cases), our goal is to make the bases on both sides of the equation the same. Once the bases are the same, we can equate the exponents and solve for the variable.

Steps: Express both sides of the equation with the same base. Equate the exponents. Solve the resulting equation.

Example 1: Solve for x: `2^x = 8` Explanation: We want to write 8 as a power of

2. We know that `8 = 2³`.

Solution: `2^x = 2³` Since the bases are the same (2), we can equate the exponents: `x = 3` Example 2: Solve for x: `3^(x+1) = 27` Explanation: We want to write 27 as a power of

3. We know that `27 = 3³`.

Solution: `3^(x+1) = 3³` Since the bases are the same (3), we can equate the exponents: `x + 1 = 3` `x = 3 - 1` `x = 2` Example 3: Solve for x: `4^x = 8` Explanation: Both 4 and 8 can be expressed as powers of 2. `4 = 2²` and `8 = 2³`.

Solution: `(2²)^x = 2³` `2^(2x) = 2³` Equate the exponents: `2x = 3` `x = 3/2` 2.4 Laws of Exponents (Review and Application) Remember and apply the laws of exponents when simplifying expressions: Product of powers: `a^m a^n = a^(m+n)` Quotient of powers: `a^m / a^n = a^(m-n)` Power of a power: `(a^m)^n = a^(mn)` Power of a product: `(ab)^n = a^n b^n` Power of a quotient: `(a/b)^n = a^n / b^n` Zero exponent: `a^0 = 1` (where a ≠ 0)

Negative exponent: `a^(-n) = 1/a^n` Guided Practice (With Solutions)

Question 1: Simplify `(x^(1/2) * y^(2/3))^6` Solution: `(x^(1/2) y^(2/3))^6 = x^(1/2 6) y^(2/3 6) = x^3 * y^4`

Commentary: We used the "power of a product" rule and then the "power of a power" rule.

Question 2: Simplify `√(9a^4b^6)` Solution: `√(9a^4b^6) = (9a^4b^6)^(1/2) = 9^(1/2) a^(4 1/2) b^(6 1/2) = 3a^2b^3`

Commentary: First, we converted from radical to exponential form. Then, we applied the power to each factor and simplified.

Question 3: Solve for x: `5^(2x-1) = 125` Solution: `5^(2x-1) = 125 = 5^3` `2x - 1 = 3` `2x = 4` `x = 2`

Commentary: We expressed 125 as a power of 5, then equated the exponents and solved for x.

Question 4: Simplify `(16x^8y^4)^(3/4)` Solution: `(16x^8y^4)^(3/4) = 16^(3/4) x^(8 3/4) y^(4 3/4) = (⁴√16)³ x^6 y^3 = 2³ x^6 y^3 = 8x^6y^3`

Commentary: We applied the exponent to each factor. Remember to evaluate the numerical part 16^(3/4). Independent Practice (Questions Only)

Simplify: `(a^(2/5) * b^(1/2))^(10)` Simplify: `³√(27x^6y^9)` Simplify: `(81a^4b^8)^(1/4)` Solve for x: `4^(3x) = 64` Solve for x: `9^(x-1) = 3` Solve for x: `(1/2)^(x) = 8` Simplify: `(x^(3/2) y^(-1/4)) / (x^(1/2) y^(1/4))` Simplify: `(25a^(6)b^(-2))^(1/2)` Solve for x: `2^(x+2) + 2^x = 20` Simplify: `(a^(1/2) + b^(1/2))(a^(1/2) - b^(1/2))`