Hello, and welcome to your mathematics lesson. Today, we are diving into Chapter Two: Exponents, also known as Powers. By the end of this lesson, you will understand how to work with positive and negative exponents, master the fundamental laws that govern them, and learn how to simplify complex expressions involving powers.
Let us begin with a quick review of what you already know. When we write a number raised to a power, we are using shorthand for repeated multiplication. If x is any real number and n is a positive integer, then xⁿ means x multiplied by itself n times. Here, x is called the base, and n is called the exponent, or index, or power.
We read xⁿ as x raised to the power n. For example, 2²⁵ means 2 multiplied by itself 25 times. The base is 2, and the exponent is 25.
Now, what happens when the exponent is negative? Consider 1/2⁵. We can write this as 2⁻⁵. So a negative exponent simply means the reciprocal. If the base is 2 and the exponent is negative 5, the number equals 2⁻⁵, which is 1/2⁵, or 1/32.
Now let us explore the three fundamental laws of exponents for integral powers.
First, the Product Law. When you multiply two powers with the same base, you add their exponents. The Product Law states: aᵐ × aⁿ = aᵐ⁺ⁿ, where a is any non-zero number and m and n are integers. For instance, 3³ × 3⁵ equals 3³⁺⁵, which is 3⁸. Similarly, 2⁻⁵ × 2⁸ becomes 2⁻⁵⁺⁸, giving us 2³ or 8.
Second, the Quotient Law. When you divide two powers with the same base, you subtract their exponents.
The Quotient Law states: aᵐ/aⁿ = aᵐ⁻ⁿ when m is greater than n, and a is non-zero. If n is greater than m, then aᵐ/aⁿ = 1/aⁿ⁻ᵐ. For example, 2¹²/2⁷ equals 2¹²⁻⁷, which is 2⁵. And 2⁶/2¹³ equals 1/2¹³⁻⁶, which is 1/2⁷.
Third, the Power Law. When you raise a power to another power, you multiply the exponents.
The Power Law states: (aᵐ)ⁿ = aᵐⁿ, where a is non-zero and m and n are integers. For example, (3⁵)² equals 3⁵ˣ², which is 3¹⁰ or 59049. Similarly, (5⁻³)⁻² becomes 5⁻³ˣ⁻², giving us 5⁶.
Let us now consider what happens when the base is negative. Look at these patterns. (-2)³ equals negative 8. (-2)⁴ equals positive 16. (-2)⁵ equals negative 32. (-2)⁶ equals positive 64.
Do you see the pattern? When the exponent is even, the result is positive. When the exponent is odd, the result is negative.
In general, (-a)ⁿ equals aⁿ if n is even, and equals -aⁿ if n is odd.
Now we turn to negative integral exponents in more detail. For any non-zero rational number a, we define: a⁻ⁿ equals 1/aⁿ, where n is a positive integer. And conversely, aⁿ equals 1/a⁻ⁿ. This means aⁿ and a⁻ⁿ are reciprocals of each other.
For example, 5⁻³ equals 1/5³. And (2/3)⁻⁵ equals (3/2)⁵. Notice how the fraction flips when the exponent is negative.
Let us work through an example together. Evaluate (-2/3)⁻⁴. First, we take the reciprocal of the base and make the exponent positive: this becomes (-3/2)⁴. Since the exponent 4 is even, the negative sign disappears, giving (3/2)⁴. This equals 3⁴/2⁴, which is 81/16.
We now move to additional important properties of exponents.
First, the power of a product. (a × b)ⁿ equals aⁿ × bⁿ. For example, (a⁵ × b⁻³)⁴ becomes a²⁰ × b⁻¹².
Second, the power of a quotient. (a/b)ⁿ equals aⁿ/bⁿ. For instance, (a⁻³/b⁴)⁶ equals a⁻¹⁸/b²⁴.
Third, let us recall the relationship between negative and positive exponents. a⁻ᵐ equals 1/aᵐ, and 1/a⁻ᵐ equals aᵐ, provided a is not zero.
Fourth, we connect exponents with roots. The nth root of a can be written as a¹/ⁿ, where n is a positive integer greater than 1. And the nth root of aᵐ equals aᵐ/ⁿ. For example, the square root of 5 is 5½, and the sixth root of 5⁷ is 5⁷/⁶.
Fifth, and crucially, any non-zero number raised to the power zero equals one.
This is the Zero Exponent Law: a⁰ equals 1, for any a not equal to zero. This follows from the Quotient Law: aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰ = 1. So 5⁰ equals 1, (-8)⁰ equals 1, and even (2⁻⁵)⁰ equals 1.
Be very careful with notation here. (-3)⁰ equals 1, but -3⁰ equals negative 1. The brackets matter enormously.
Let us apply these laws to evaluate some expressions.
Consider 4³/² × 125⁻²/³.
First, rewrite 4 as 2² and 125 as 5³. This becomes (2²)³/² × (5³)⁻²/³. Using the Power Law, multiply the exponents: this gives 2³ × 5⁻². This equals 8/5², which is 8/25.
Here is another example.
Evaluate (8/27)²/³ ÷ 32⁻²/⁵. First, note that 8/27 equals (2/3)³, and 32 equals 2⁵. So we have ((2/3)³)²/³ ÷ (2⁵)⁻²/⁵. This simplifies to (2/3)² ÷ 2⁻². Which equals 4/9 × 4, giving us 16/9 or 1 7/9.
One more example involving negative bases and zero exponents. Evaluate -2⁴ − (√3)⁰ × (-2)⁶ ÷ 4. Step by step: −2⁴ equals negative 16. (√3)⁰ equals 1. (-2)⁶ equals 2⁶ which is 64. And 4 equals 2². So the final result is negative 32.
Let us see how to solve equations involving exponents.
Solve for x: 9ˣ⁻¹/3 = 27. First, express everything with base 3. 9 is 3², and 27 is 3³. So we have 3³ˣ⁻¹/3² = 3³. Using the Quotient Law, this becomes 3³ˣ⁻¹⁻² = 3³. So 3³ˣ⁻³ = 3³. Since the bases are equal, the exponents must be equal: 3x - 3 = 3. Therefore 3x = 6, and x = 2.
Here is another equation to solve. Solve 25ⁿ⁻¹ + 100 = 5²ⁿ⁻¹, find n. Rewrite 25 as 5². So (5²)ⁿ⁻¹ + 100 = 5²ⁿ⁻¹. This gives 5²ⁿ⁻² + 100 = 5²ⁿ⁻¹. Rearranging: 100 equals 5²ⁿ⁻¹ - 5²ⁿ⁻². Factor out 5²ⁿ: this becomes 5²ⁿ(5⁻¹ − 5⁻²) equals 100. Now 5⁻¹ - 5⁻² equals 1/5 - 1/25, which is 4/25. So 5²ⁿ × 4/25 = 100. Therefore 5²ⁿ = 100 × 25/4, which is 625. Since 625 equals 5⁴, we have 2n = 4, so n = 2.
Let us recap the key takeaways from this lesson.
First, the three fundamental laws: the Product Law tells us that when multiplying powers with the same base, we add exponents. The Quotient Law tells us that when dividing powers with the same base, we subtract exponents. And the Power Law tells us that when raising a power to another power, we multiply the exponents.
Second, negative exponents represent reciprocals: a⁻ⁿ equals 1/aⁿ.
Third, any non-zero number raised to the power zero equals one.
Fourth, when the base is negative, an even exponent gives a positive result, while an odd exponent gives a negative result.
Fifth, fractional exponents connect to roots: aᵐ/ⁿ equals the nth root of aᵐ, where m and n are positive integers and n is greater than 1.
And finally, powers of products and quotients distribute: (ab)ⁿ equals aⁿbⁿ, and (a/b)ⁿ equals aⁿ/bⁿ.
That brings us to the end of this lesson on Exponents and Powers. You have learned the essential laws and techniques for working with exponents, both positive and negative, integral and fractional. Practice applying these laws systematically, and you will find that even complex expressions become manageable. Keep exploring, keep practicing, and enjoy your journey through mathematics. Until next time, goodbye.