Exponents with Negative Fractional Bases

Exponents play a fundamental role in mathematics, enabling us to express repeated multiplication in a concise and efficient manner.

As Grade 7 students embark on their mathematical journey, they encounter a fascinating twist: negative fractional bases. These seemingly complex expressions involve negative numbers raised to fractional powers.

Table of Contents:

• Exponents with Negative Fractional Bases
• Formula
• Solved Examples
• FAQs

Exponents with Negative Fractional Bases

Negative fractional exponents might appear intimidating at first, but they hold fascinating mathematical properties. These exponents involve raising a negative number to a fraction, such as (-3)^(1/2) or (-2)^(2/3).

In simple terms, negative fractional exponents introduce the concept of taking roots of negative numbers. By exploring these exponents, Grade 7 students will gain a deeper understanding of how negative numbers can be raised to fractional powers and the unique outcomes that result.

Unleashing the Power: Properties of Negative Fractional Exponents

To navigate the world of exponents with negative fractional bases, it’s important to grasp their properties. Negative fractional exponents exhibit specific rules and behaviors. Students will learn about the power rule, which states that raising a negative base to a negative fractional exponent can be rewritten as its reciprocal with a positive exponent. They will also explore the concept of even and odd roots, understanding why certain negative fractional exponents yield positive results while others result in negative values.

Mastering the Calculations: Evaluating Expressions with Negative Fractional Exponents

When it comes to evaluating expressions with negative fractional exponents, Grade 7 students will develop the skills to simplify and calculate their values. They will learn techniques such as converting negative fractional exponents to radical form and vice versa. By practicing these calculations, students will gain confidence in dealing with negative fractional exponents and perform accurate evaluations of complex mathematical expressions.

Real-World Applications: Harnessing the Power of Negative Fractional Exponents

While it may seem abstract, understanding exponents with negative fractional bases has practical applications in the real world. Students will explore examples from fields such as physics, engineering, and finance. They will discover how negative fractional exponents are used to model exponential decay, growth rates, fractional dimensions, and other phenomena. By connecting these concepts to real-life scenarios, Grade 7 students will recognize the relevance and power of negative fractional exponents beyond the confines of the classroom.

Formulas

Power Rule for Negative Fractional Exponents:

For any nonzero number ‘a’ and a negative fractional exponent ‘m/n’ (where ‘m’ and ‘n’ are positive integers):

a^(-m/n) = 1 / (a^(m/n))

Converting Negative Fractional Exponents to Radical Form:

For any nonzero number ‘a’ and a negative fractional exponent ‘m/n’ (where ‘m’ and ‘n’ are positive integers):

a^(-m/n) = 1 / (n-th root of (a^m))

Even Root of a Negative Number:

For any positive even integer ‘n’ and a negative number ‘a’:

n-th root of (-a) = – (n-th root of a)

Odd Root of a Negative Number:

For any positive odd integer ‘n’ and a negative number ‘a’:

n-th root of (-a) = – (n-th root of a)

Solved Examples

Example 1: Evaluate (-8)^(2/3).

Solution:

Using the formula for converting negative fractional exponents to radical form, we have:

(-8)^(2/3) = 1 / (3rd root of (-8))^2

Since the exponent is positive after converting to radical form, we can simplify further:

(-8)^(2/3) = 1 / (3rd root of 8)^2

Now, we can evaluate the 3rd root of 8, which is 2:

(-8)^(2/3) = 1 / 2^2

(-8)^(2/3) = 1 / 4

(-8)^(2/3) = 1/4

Therefore, (-8)^(2/3) equals 1/4.

Example 2: Evaluate (-27)^(3/5).

Solution:

Using the formula for converting negative fractional exponents to radical form, we have:

(-27)^(3/5) = 1 / (5th root of (-27))^3

Since the exponent is positive after converting to radical form, we can simplify further:

(-27)^(3/5) = 1 / (5th root of 27)^3

Now, we can evaluate the 5th root of 27, which is 3:

(-27)^(3/5) = 1 / 3^3

(-27)^(3/5) = 1 / 27

(-27)^(3/5) = 1/27

Therefore, (-27)^(3/5) equals 1/27.

Example 3: Evaluate (-16)^(1/4).

Solution:

Using the formula for converting negative fractional exponents to radical form, we have:

(-16)^(1/4) = 1 / (4th root of (-16))^1

Since the exponent is positive after converting to radical form, we can simplify further:

(-16)^(1/4) = 1 / (4th root of 16)

Now, we can evaluate the 4th root of 16, which is 2:

(-16)^(1/4) = 1 / 2

(-16)^(1/4) = 1/2

Therefore, (-16)^(1/4) equals 1/2.

FAQs

Grade 7 Science Worksheets

• Elements and Compounds
• Solar Energy
• Photosynthesis
• Electricity and Magnetism
• Law of conservation of energy
• Periodic table
• Properties of Matter
• Waves
• Energy Resources 
• Weather and Climate
• Immune, Circulatory and Digestive Systems
• Organs in Multi-cellular Organism
• Sedimentary, Igneous, and Metamorphic Rocks
• Structure of the Earth
• Law of Conservation of Mass
• Physical and Chemical Changes
• Scientific Method
• Human Digestive System
• Environmental Science
• Renewable and Non-renewable energy Resources
• Characteristics of Living Organisms
• Life Science
• Earth and Space Science
• Solar Eclipse
• Heat Technology
• Newton’s Laws of Motions
• Physical Science
• Tools, Measurement and SI Units
• Earth Atmosphere
• Interactions of Living things
• The Earth Ecosystem
• Organelles in Plant and Animal cells
• Layers of the Earth
• Cycles in Nature

Grade 7 Math Worksheets

• Fractions
• Linear equations word problems
• Statistics
• Properties of Parallel Line
• Finding slope from an equation
• Identifying Quadrilaterals
• Percent Change
• Properties of addition and multiplication
• Pythagorean Theorem
• Solving two step inequalities
• Symmetry
• Fractions to Decimals (New)
• Whole Number Exponents with Integer Bases (New)
• Adding and Subtracting Fractions (New)
• Integer Addition and Subtraction (New)
• Dividing Mixed Numbers (New)
• Basics of Coordinate Geometry (New)