27 Divided By (3/4) Square is a mathematical expression that involves understanding division, fractions, and exponents. Whether you're a student working through algebra problems or someone interested in mathematical concepts, grasping how to evaluate this expression is essential. In this comprehensive article, we'll explore the problem step-by-step, explain relevant mathematical principles, and provide helpful tips to understand and solve similar expressions. By the end, you'll have a clear understanding of how to evaluate 27 divided by (3/4) squared and appreciate the underlying mathematical framework.
Before diving into the calculations, it's important to dissect the expression:
- 27: a whole number, also known as an integer.
- Divided by (/): a division operation.
- (3/4): a fraction, representing three parts out of four.
- Squared (²): an exponent indicating that the fraction is multiplied by itself.
The expression can be read as: "Twenty-seven divided by the square of three-fourths." To evaluate it accurately, we need to understand how to handle division, fractions, and exponents.
Let's break down the calculation process systematically.
Step 1: Understand the Squaring Operation
Squaring a fraction involves multiplying the fraction by itself:
\[
\left(\frac{3}{4}\right)^2 = \frac{3}{4} \times \frac{3}{4}
\]
When multiplying fractions, multiply the numerators together and the denominators together:
\[
\frac{3 \times 3}{4 \times 4} = \frac{9}{16}
\]
So, (3/4) squared equals 9/16.
Step 2: Rewrite the original expression
Replacing the squared part with its simplified form:
\[
27 \div \frac{9}{16}
\]
Note: Dividing by a fraction is equivalent to multiplying by its reciprocal.
Step 3: Multiply by the reciprocal
Using the reciprocal of 9/16, which is 16/9:
\[
27 \div \frac{9}{16} = 27 \times \frac{16}{9}
\]
Now, perform the multiplication:
\[
27 \times \frac{16}{9}
\]
Step 4: Simplify the multiplication
Express 27 as a fraction:
\[
\frac{27}{1} \times \frac{16}{9}
\]
Multiply numerators and denominators:
\[
\frac{27 \times 16}{1 \times 9} = \frac{432}{9}
\]
Simplify numerator and denominator:
\[
\frac{432}{9} = 48
\]
Result: The value of 27 divided by (3/4) squared is 48.
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Understanding this problem involves several fundamental mathematical concepts:
Key Concepts:
- Division of whole numbers by fractions: Dividing by a fraction is equivalent to multiplying by its reciprocal.
- Exponentiation of fractions: Squaring a fraction involves multiplying it by itself.
- Multiplying fractions: Multiply numerators and denominators directly.
- Simplification: Always simplify your results to their lowest terms for clarity and accuracy.
Application to Other Expressions
The approach used here can be applied universally:
- When dividing a number by a fraction, invert the fraction and multiply.
- When dealing with powers of fractions, multiply numerator and denominator separately.
- Simplify the resulting fraction before performing further calculations.
To reinforce your understanding, consider practicing with similar problems:
- Calculate 50 divided by (2/3) squared.
- Evaluate 15 divided by (5/8) squared.
- Find the value of 100 divided by (7/10) squared.
Solutions:
- \(50 \div \left(\frac{2}{3}\right)^2 = 50 \times \left(\frac{3}{2}\right)^2 = 50 \times \frac{9}{4} = \frac{50 \times 9}{4} = \frac{450}{4} = 112.5\)
- \(15 \div \left(\frac{5}{8}\right)^2 = 15 \times \left(\frac{8}{5}\right)^2 = 15 \times \frac{64}{25} = \frac{15 \times 64}{25} = \frac{960}{25} = 38.4\)
- \(100 \div \left(\frac{7}{10}\right)^2 = 100 \times \left(\frac{10}{7}\right)^2 = 100 \times \frac{100}{49} = \frac{100 \times 100}{49} = \frac{10,000}{49} \approx 204.08\)
- Always handle exponents before division to simplify calculations.
- Remember that dividing by a fraction is equivalent to multiplying by its reciprocal.
- Simplify fractions at every step to avoid errors.
- Use prime factorization to simplify large numbers when possible.
Evaluating 27 divided by (3/4) squared involves understanding how to handle exponents of fractions and division of whole numbers by fractions. The key steps are:
- Square the fraction: \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\).
- Rewrite the division as multiplication by the reciprocal: \(27 \div \frac{9}{16} = 27 \times \frac{16}{9}\).
- Simplify the multiplication: \(27 \times \frac{16}{9} = \frac{432}{9} = 48\).
The final answer is 48. This process illustrates fundamental algebraic principles that are applicable to a wide range of mathematical problems involving division, fractions, and exponents.
By mastering these steps, you'll be better equipped to tackle more complex expressions and deepen your understanding of mathematical relationships. Practice regularly with similar problems to build confidence and improve your problem-solving skills in mathematics.
