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A cube root is a mathematical operation that finds a number that, when multiplied by itself three times, gives the original number. It is the inverse operation of cubing a number. The symbol for the cube root is \( \sqrt[3]{\ } \). For example, the cube root of 8 is 2, because 2 multiplied by itself three times results in 8.
\[ 2 \times 2 \times 2 = 8 \]
In algebra, cube roots can be used to simplify expressions involving variables raised to powers. Consider an expression like \( \sqrt[3]{y^5} \). To find the cube root, we think of the exponent of \( y \) as being divided by 3. This is equivalent to raising \( y^5 \) to the power of \( \frac{1}{3} \). This brings us to the idea of fractional exponents, a helpful concept in dealing with roots.

Exponents are powerful notations in mathematics that indicate how many times a number, the base, is multiplied by itself. Understanding the properties of exponents helps in simplifying complex expressions. Some fundamental properties include:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \((a^m)^n = a^{m \cdot n} \)

When dealing with expressions like \( (y^5)^{\frac{1}{3}} \), the property "power of a power" is especially useful. It helps us rearrange and simplify expressions by multiplying the exponents. For \((y^5)^{\frac{1}{3}}\), we multiply the 5 by \( \frac{1}{3} \) to get \( y^{\frac{5}{3}} \). This makes the expression much easier to manage and understand.

Fractional exponents are another way to represent roots. They combine the notions of powers and roots into a single expression. The general form is \( a^{\frac{m}{n}} \), which is equivalent to the \( n\)-th root of \( a^m \), or \( \sqrt[n]{a^m} \). For example, \( 8^{\frac{2}{3}} \) means the cube root of \( 8^2 \), or \( \sqrt[3]{64} \), which simplifies down to 4.
In our specific problem, we simplify \( y^{\frac{5}{3}} \). Here, 5 is your numerator, indicating the power. The denominator 3 indicates the cube root. Therefore, \( y^{\frac{5}{3}} \) can be broken down as \( y^1 \times y^{\frac{2}{3}} \). Understanding fractional exponents is crucial as they provide a compact way to express both powers and roots, making calculations more concise.