Problem 95
Question
In Exercises \(95-102,\) simplify by reducing the index of the radical. $$\sqrt[4]{5^{2}}$$
Step-by-Step Solution
Verified Answer
\(\sqrt{5}\)
1Step 1: Identify the Radicand, Power and Index
Firstly, identify the elements in the expression. Here, the radicand (the number inside the radical) is 5, the power is 2, and the index (the number outside the radical) is 4.
2Step 2: Perform Division
Now, divide the power by the index. In this case, it's \(2/4 = 0.5\). By definition, the square root means the power is \(\frac{1}{2}\).
3Step 3: Apply the Power Rule of Radicals
Apply the power rule, which means that a radicand to the power of \(\frac{1}{2}\) is equal to its square root. Therefore, \(5^{0.5}\) is equivalent to \(\sqrt{5}\).
4Step 4: Final Simplification
The final step is to simplify the result, but in this case, \(\sqrt{5}\) cannot be simplified further.
Key Concepts
Radical ExpressionsIndex of a RadicalPower Rule of Radicals
Radical Expressions
Understanding radical expressions is crucial for mastering the foundations of algebra. Simply put, a radical expression includes a number or expression under the root symbol. This may look intimidating at first, but it’s really just another form of an exponent.
When you come across \(\sqrt[n]{a}\), where \(n\)is the index of the radical and \(a\) is the radicand, it’s asking you, “What number to the \(n\)th power gives us \(a\)?” For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
Often, you’ll need to simplify these expressions to make them easier to handle in calculations. This might involve finding equivalent expressions with smaller radicals or without radicals entirely. The key to simplifying is to understand and manipulate the components of the radical—namely, the radicand and the index.
When you come across \(\sqrt[n]{a}\), where \(n\)is the index of the radical and \(a\) is the radicand, it’s asking you, “What number to the \(n\)th power gives us \(a\)?” For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
Often, you’ll need to simplify these expressions to make them easier to handle in calculations. This might involve finding equivalent expressions with smaller radicals or without radicals entirely. The key to simplifying is to understand and manipulate the components of the radical—namely, the radicand and the index.
Index of a Radical
The index of a radical indicates the degree of the root you’re dealing with. In an expression like \(\sqrt[n]{a}\), the index is the small number \(n\) outside and above the radical sign. When no index is visually present, such as in \(\sqrt{a}\), it is assumed to be 2, which represents a square root.
Getting comfortable with the index is key to simplifying radical expressions. For instance, if the index is 3, you're looking for a number that, when multiplied by itself three times, gives the radicand.
When simplifying a radical expression, one approach is to divide the exponent of the radicand by the index. This can sometimes reduce the radical to a simpler form or even to an integer. For indices greater than the exponent of the radicand, as in the given exercise example, it leads to fractional exponents that represent the roots of the radicand.
Getting comfortable with the index is key to simplifying radical expressions. For instance, if the index is 3, you're looking for a number that, when multiplied by itself three times, gives the radicand.
When simplifying a radical expression, one approach is to divide the exponent of the radicand by the index. This can sometimes reduce the radical to a simpler form or even to an integer. For indices greater than the exponent of the radicand, as in the given exercise example, it leads to fractional exponents that represent the roots of the radicand.
Power Rule of Radicals
The power rule of radicals is a mathematical shortcut that helps simplify expressions involving roots. The general rule states that \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\) when \(a\) is a non-negative real number, \(m\) is the exponent, and \(n\) is the index of the radical.
For simplifying purposes, if the exponent \(m\) is divisible by the index \(n\), the radical can be removed completely by writing the radicand to the power of the quotient. This is visible in the exercise example, where \(\sqrt[4]{5^2} = 5^{\frac{2}{4}}\) simplifies to \(5^{0.5}\).
However, there are times when the exponent isn't a clean multiple of the index, and partial simplification may involve leaving a radical. But worry not; practice with these concepts makes it easier over time. Remember, mastering the power rule can dramatically speed up your problem-solving process when dealing with radicals.
For simplifying purposes, if the exponent \(m\) is divisible by the index \(n\), the radical can be removed completely by writing the radicand to the power of the quotient. This is visible in the exercise example, where \(\sqrt[4]{5^2} = 5^{\frac{2}{4}}\) simplifies to \(5^{0.5}\).
However, there are times when the exponent isn't a clean multiple of the index, and partial simplification may involve leaving a radical. But worry not; practice with these concepts makes it easier over time. Remember, mastering the power rule can dramatically speed up your problem-solving process when dealing with radicals.
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