Learn on PengiReveal Math, AcceleratedUnit 13: Irrational Numbers, Exponents, and Scientific Notation

Lesson 13-5: Explore Patterns of Exponents

In this Grade 7 lesson from Reveal Math, Accelerated, students explore patterns of exponents by investigating the zero exponent rule and the negative exponent rule, learning that any nonzero number raised to the zero power equals 1 and that a negative exponent represents the multiplicative inverse of the corresponding positive power. Students connect these rules to real-world contexts like metric system unit conversions and exponential growth through paper folding. The lesson builds fluency in evaluating and comparing expressions with zero and negative exponents, including fractional bases.

Section 1

Repeated Multiplication and Exponential Form

Property

Any repeated multiplication can be written in exponential form: a⋅a⋅a⋅…⋅aa \cdot a \cdot a \cdot \ldots \cdot a (nn factors) = ana^n.

Conversely, any exponential expression can be expanded back into repeated multiplication. The base (aa) indicates what number is being multiplied, and the exponent (nn) indicates how many times the base appears as a factor.

Examples

  • 5⋅5⋅5⋅5=545 \cdot 5 \cdot 5 \cdot 5 = 5^4 (four factors of 5)
  • x6=x⋅x⋅x⋅x⋅x⋅xx^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x (six factors of xx)
  • (−3)⋅(−3)⋅(−3)=(−3)3(-3) \cdot (-3) \cdot (-3) = (-3)^3 (three factors of -3)

Section 2

Exploring Zero and Negative Exponents

Property

As the exponent of a base decreases by 1, the value of the power is divided by the base. Following this pattern past the exponent of 1 reveals the rules for zero and negative exponents:
For every nonzero number xx, x0=1x^0 = 1.
For every nonzero number xx, x−n=1xnx^{-n} = \frac{1}{x^n}.

Examples

  • Consider the pattern for powers of 2:
23=82^3 = 8
22=4(since 8÷2=4)2^2 = 4 \quad (\text{since } 8 \div 2 = 4)
21=2(since 4÷2=2)2^1 = 2 \quad (\text{since } 4 \div 2 = 2)
20=1(since 2÷2=1)2^0 = 1 \quad (\text{since } 2 \div 2 = 1)
2−1=12(since 1÷2=12)2^{-1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})
2−2=14(since 12÷2=14)2^{-2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})

Notice that 2−22^{-2} is exactly the same as 122\frac{1}{2^2}.

  • Simplify x−4y2\frac{x^{-4}}{y^2}: Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in 1x4y2\frac{1}{x^4 y^2}.

Explanation

Zero and negative exponents aren't magic; they are just the logical continuation of a mathematical pattern! Every time an exponent drops by one, you divide by the base. This proves why any non-zero number to the power of zero is exactly 1. It also shows that a negative exponent is basically a "flip-it" command: it tells you to take the reciprocal of the base and make the exponent positive.

Section 3

Evaluating Exponents with Negative Bases

Property

The placement of parentheses completely changes the meaning and the result of an expression with a negative base.
In (−a)n(-a)^n, the base is −a-a and the entire negative number is multiplied nn times.
In −an-a^n, the base is just aa; the exponent is applied first, and then the negative sign is attached to the final result.

Examples

  • To simplify (−3)2(-3)^2, the base is -3. You calculate (−3)(−3)=9(-3)(-3) = 9.
  • To simplify −32-3^2, the base is 3. You calculate 32=93^2 = 9 first, and then take the opposite, giving -9.
  • Evaluate k2−kk^2 - k for k=−6k = -6:

Substitute with parentheses: (−6)2−(−6)=36+6=42(-6)^2 - (-6) = 36 + 6 = 42.

Explanation

Parentheses act like a protective force field! When you write (−5)2(-5)^2, you're telling the math world to square the entire thing inside, negative sign and all, resulting in a positive 25. But without that force field, −52-5^2 means you only square the 5, and the negative sign just waits outside to get tacked on at the very end. Always use parentheses when substituting negative numbers!

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Repeated Multiplication and Exponential Form

Property

Any repeated multiplication can be written in exponential form: a⋅a⋅a⋅…⋅aa \cdot a \cdot a \cdot \ldots \cdot a (nn factors) = ana^n.

Conversely, any exponential expression can be expanded back into repeated multiplication. The base (aa) indicates what number is being multiplied, and the exponent (nn) indicates how many times the base appears as a factor.

Examples

  • 5⋅5⋅5⋅5=545 \cdot 5 \cdot 5 \cdot 5 = 5^4 (four factors of 5)
  • x6=x⋅x⋅x⋅x⋅x⋅xx^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x (six factors of xx)
  • (−3)⋅(−3)⋅(−3)=(−3)3(-3) \cdot (-3) \cdot (-3) = (-3)^3 (three factors of -3)

Section 2

Exploring Zero and Negative Exponents

Property

As the exponent of a base decreases by 1, the value of the power is divided by the base. Following this pattern past the exponent of 1 reveals the rules for zero and negative exponents:
For every nonzero number xx, x0=1x^0 = 1.
For every nonzero number xx, x−n=1xnx^{-n} = \frac{1}{x^n}.

Examples

  • Consider the pattern for powers of 2:
23=82^3 = 8
22=4(since 8÷2=4)2^2 = 4 \quad (\text{since } 8 \div 2 = 4)
21=2(since 4÷2=2)2^1 = 2 \quad (\text{since } 4 \div 2 = 2)
20=1(since 2÷2=1)2^0 = 1 \quad (\text{since } 2 \div 2 = 1)
2−1=12(since 1÷2=12)2^{-1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})
2−2=14(since 12÷2=14)2^{-2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})

Notice that 2−22^{-2} is exactly the same as 122\frac{1}{2^2}.

  • Simplify x−4y2\frac{x^{-4}}{y^2}: Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in 1x4y2\frac{1}{x^4 y^2}.

Explanation

Zero and negative exponents aren't magic; they are just the logical continuation of a mathematical pattern! Every time an exponent drops by one, you divide by the base. This proves why any non-zero number to the power of zero is exactly 1. It also shows that a negative exponent is basically a "flip-it" command: it tells you to take the reciprocal of the base and make the exponent positive.

Section 3

Evaluating Exponents with Negative Bases

Property

The placement of parentheses completely changes the meaning and the result of an expression with a negative base.
In (−a)n(-a)^n, the base is −a-a and the entire negative number is multiplied nn times.
In −an-a^n, the base is just aa; the exponent is applied first, and then the negative sign is attached to the final result.

Examples

  • To simplify (−3)2(-3)^2, the base is -3. You calculate (−3)(−3)=9(-3)(-3) = 9.
  • To simplify −32-3^2, the base is 3. You calculate 32=93^2 = 9 first, and then take the opposite, giving -9.
  • Evaluate k2−kk^2 - k for k=−6k = -6:

Substitute with parentheses: (−6)2−(−6)=36+6=42(-6)^2 - (-6) = 36 + 6 = 42.

Explanation

Parentheses act like a protective force field! When you write (−5)2(-5)^2, you're telling the math world to square the entire thing inside, negative sign and all, resulting in a positive 25. But without that force field, −52-5^2 means you only square the 5, and the negative sign just waits outside to get tacked on at the very end. Always use parentheses when substituting negative numbers!