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Chapter 7: Problem 17

Write each expression in radical form. $$y^{1.2}$$

Short Answer

Expert verified
The expression \(y^{1.2}\) in radical form is \(\sqrt[5]{y^6}\).

Step by step solution

01

Convert Decimal to Fraction

To begin with, convert the decimal exponent to a fraction. The decimal 1.2 can be expressed as \(12/10\). By simplifying the fraction, it becomes \(6/5\). So, \(y^{1.2}\) can be rewritten as \(y^{6/5}\).
02

Translate to Radical Form

Next, translate the expression into radical form. In the exponent \(6/5\), the numerator 6 represents the power of the base number \(y\), and the denominator 5 signifies the root. Together they form a radical expression. The fractional exponent \(y^{6/5}\) translates to the radical expression \(\sqrt[5]{y^6}\).

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Key Concepts

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

Fractional Exponents
Fractional exponents can seem tricky at first, but they're quite manageable with a little understanding. The idea is simple: it's about combining powers and roots in one expression. When you see an exponent like \(y^{1.2}\), it means there's both a power and a root involved. Let's break it down further.

Fractional exponents such as \(y^{6/5}\) tell us two things:
  • The denominator, 5, is the root. This means we're dealing with the 5th root of something.
  • The numerator, 6, is the power. This means the base, \(y\), is raised to the 6th power.
Putting it all together, \(y^{6/5}\) is a combination of a power and a root, resulting in \(\sqrt[5]{y^6}\). This is the translated radical form. Always remember: denominator outside, numerator inside. This mental check helps you avoid common mistakes when moving between fractional exponents and radicals.
Decimal to Fraction Conversion
Converting decimals to fractions is an essential skill in math, especially when working with exponents. To convert a decimal like 1.2 into a fraction, remember these steps:

  • Count the number of decimal places. Here, 1.2 has one decimal place.
  • Write the number without the decimal as the numerator (12) and a denominator of 10 (since 1 decimal place means tenths).
Thus, 1.2 becomes \(\frac{12}{10}\).

Next, it's all about simplifying. Simplifying is crucial because it gives the smallest terms possible, making it easier to work with. For \(\frac{12}{10}\), divide both the numerator and denominator by 2, the greatest common divisor, which gives \(\frac{6}{5}\). Converting to a simplified fraction ensures your math stays simple and accurate.
Simplifying Fractions
Simplifying fractions is a key math skill that helps with clarity and efficiency in calculations. Simplification means reducing the fraction to its smallest terms.

For instance, if we start with \(\frac{12}{10}\), the greatest common divisor (GCD) helps us find the smallest version of this fraction. In this case, the GCD of 12 and 10 is 2. By dividing both the numerator and denominator by their GCD, the fraction reduces to \(\frac{6}{5}\).

  • This process involves finding common factors for both numbers in the fraction.
  • Keep dividing by the GCD until you can't reduce the fraction further.
The result is a fraction that's simpler and easier to handle, especially when plugging back into expressions or equations. Simplifying is like tidying up your math: it keeps everything neat and manageable!

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