Introduction
Fractions and powers are two mathematical concepts that are essential in everyday life, particularly when dealing with numbers. Fractions are used to represent parts of a whole, while powers denote repeated multiplication. When we combine fractions with powers, we get a new concept known as fractions with powers. In this blog post, we will explore fractions with powers, their properties, and some examples.
What are Fractions with Powers?
Fractions with powers are expressions that involve both fractions and powers. For example, 3/4^2 is an instance of a fraction with power. In this case, the denominator (4^2) represents repeated multiplication, while the numerator (3) represents a part of the whole. Fractions with powers are commonly used in algebraic expressions, and they have several properties that we need to understand.
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Properties of Fractions with Powers
Power of a Fraction
When a fraction is raised to a power, the numerator and denominator are raised to that power. For instance, (3/4)^2 = 3^2/4^2 = 9/16. In this case, both the numerator and denominator have been raised to the power of 2.
Fraction of a Power
When a power is expressed as a fraction, the numerator and denominator of the fraction are raised to that power. For example, (3^2/4)^3 = (3^2)^3/4^3 = 27/64. In this case, both the numerator and denominator of the fraction have been raised to the power of 3.
Multiplication of Fractions with Powers
When multiplying fractions with powers, we multiply the numerators and denominators separately. For example, (3/4^2) x (2/5^3) = (3×2)/(4^2×5^3) = 6/500. In this case, we have multiplied the numerators 3 and 2, and the denominators 4^2 and 5^3 separately.
Division of Fractions with Powers
When dividing fractions with powers, we invert the second fraction and multiply it by the first fraction. For instance, (3/4^2) ÷ (2/5^3) = (3/4^2) x (5^3/2) = (3×5^3)/(4^2×2) = 375/32. In this case, we have inverted the second fraction (2/5^3) to (5^3/2) and then multiplied it by the first fraction (3/4^2).
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Examples of Fractions with Powers
Example 1: Simplify (2/3)^4
Solution:
(2/3)^4 = 2^4/3^4 = 16/81
Example 2: Evaluate (1/2^3) x (3/4^2)
Solution:
(1/2^3) x (3/4^2) = (1×3)/(2^3×4^2) = 3/128
Example 3: Simplify (3^2/4)^3
Solution:
(3^2/4)^3 = (3^2)^3/4^3 = 27/64
Example 4: Evaluate (2/5^2) ÷ (4/3^3)
Solution:
(2/5^2) ÷ (4/3^3) = (2/5^2) x (3^3/4) = (2×3^3)/(5^2×4) = 27/1000
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Summary
Fractions with powers are expressions that involve fractions and powers, and they are commonly used in mathematical calculations and algebraic expressions. Understanding the properties of fractions with powers is crucial in simplifying these expressions and solving equations that involve them.
From the properties discussed above, we can see that when we raise a fraction to a power, we raise both the numerator and denominator to that power. Similarly, when a power is expressed as a fraction, we raise both the numerator and denominator to that power. When we multiply or divide fractions with powers, we multiply or divide the numerators and denominators separately or invert the second fraction and multiply it by the first fraction.
It is important to note that when simplifying fractions with powers, we need to apply the order of operations correctly to get the correct answer. Additionally, we need to be careful with negative exponents as they can change the value of the expression.
In closing, fractions with powers are a crucial concept in mathematics, and understanding their properties is essential in solving problems that involve them. By following the order of operations and being careful with negative exponents, we can simplify fractions with powers and solve equations that involve them effectively.