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Chapter 10: Problem 12

Write in radical form and evaluate. \(\left(\frac{4}{9}\right)^{1 / 2}\)

Short Answer

Expert verified
The given expression \(\left(\frac{4}{9}\right)^{1/2}\) can be written in radical form as \(\sqrt{\frac{4}{9}}\), which simplifies to \(\frac{2}{3}\).

Step by step solution

01

Write the expression in radical form

The expression \(\left(\frac{4}{9}\right)^{1/2}\) can be written as a square root as follows: \[\sqrt{\frac{4}{9}}\]
02

Simplify the radical expression

We can simplify the expression by taking the square root of both the numerator and the denominator individually: \[\sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}}\] Now, we can simplify the square roots: \[\frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3}\] So, the simplified radical expression is \(\frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Radical Expressions
A radical expression involves an expression that uses a root sign, commonly known as the radical symbol (鈭�). It often includes numbers or variables inside the symbol, which signifies the root to be taken. For example, when you see \( \sqrt{9} \), it means you're looking for the number that, when multiplied by itself, gives back 9. In this case, that number is 3. These expressions might look tricky at first, but don't worry, they're not as daunting as they seem!
  • Radicals are akin to fractions in arithmetic. They often simplify into easier forms.
  • Squares and square roots work together closely. Knowledge of one enhances understanding of the other.
  • The radical sign says 'find a number that, raised to this power, equals what's inside.' For square roots, this power is 2.
Learning how to simplify radical expressions can make calculations easier and more intuitive by breaking them into components. This skill also helps in encountering more complex algebraic expressions.
Explaining Fractional Exponents
Fractional exponents may look a bit funky at first glance, but they're really just another way to show roots. The nifty thing is they can make some calculations more straightforward. For instance, the expression \( \left(\frac{4}{9}\right)^{1/2} \) involves a fractional exponent, which tells us to take a square root. The denominator of the fraction (in this case, 2) represents the root you are taking.
  • The expression \( a^{1/n} \) is the same as \( \sqrt[n]{a} \), meaning you take the nth root of \( a \).
  • Fractional exponents allow combining properties of both powers and roots.
  • They are particularly helpful in algebra, where manipulation of powers is frequent.
Consider them like normal exponents. Multiplication, division, or raising to another power can still apply, only these operations may now involve fractions and whole numbers simultaneously.
The Power of Square Roots
Square roots are one of those concepts that show up a lot in math, especially when dealing with quadratic equations or geometry. Essentially, finding a square root is asking 'what number, multiplied by itself, gives this value?' In the exercise given, \( \sqrt{\frac{4}{9}} \), we look for a number that will multiply by itself to result in \( \frac{4}{9} \).
  • Square roots are inverses of squaring a number.
  • They simplify equations and expressions by offering a straightforward representation.
  • Squares (\(n^2\)) and square roots (\(\sqrt{n}\)) are fundamental to solving real-world problems involving area or quadratic equations.
Square roots provide valuable ways to express relationships between numbers, particularly in terms of symmetry and geometric calculations.

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