Problem 112 - Intermediate Algebra

Question

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt{x^{7} y^{8}}=\sqrt{x^{7}} \cdot \sqrt{y^{8}} $$

Step-by-Step Solution

Answer

True

Step 1: Understand the Radicals

The given equation is \( \sqrt{x^{7} y^{8}}=\sqrt{x^{7}} \cdot \sqrt{y^{8}} \). We need to determine if this equality is true under the properties of radicals and real numbers.

Step 2: Apply the Property of Product of Radicals

We use the property that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) if both \( a \) and \( b \) are non-negative. In this case, \( a = x^{7} \) and \( b = y^{8} \). Given that the radicals represent nonzero real numbers, they must be positive.

Step 3: Simplify Each Radical Expression

Simplify \( \sqrt{x^{7}} \) and \( \sqrt{y^{8}} \). \( \sqrt{x^{7}} = x^{7/2} \) and \( \sqrt{y^{8}} = y^{8/2} = y^{4} \). Both expressions are real and defined since \( x \) and \( y \) are nonzero real numbers.

Step 4: Confirm the Original Equation

Simplify the original radical: \( \sqrt{x^{7}y^{8}} = (x^{7}y^{8})^{1/2} = x^{7/2}y^{8/2} = x^{7/2}y^{4} \). This is the same result as multiplying the separate radicals: \( x^{7/2} \cdot y^{4} \).

Step 5: Conclusion

We conclude that \( \sqrt{x^{7} y^{8}} = \sqrt{x^{7}} \cdot \sqrt{y^{8}} \) holds true for all nonzero real numbers \( x \) and \( y \). The property of the product of radicals was correctly applied, making the expression valid.

Key Concepts

Properties of Radicals

Radicals possess many properties that are useful in simplifying and calculating various mathematical expressions. One such important property is related to the product of radicals. The rule states that the square root of a product is equal to the product of the square roots of each factor. Mathematically, it means if you have two non-negative numbers, say \( a \) and \( b \), then \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This property is very helpful when dealing with complex radical expressions.

Understanding and applying these properties allow for easier manipulation and simplification of broader mathematical expressions. Other useful properties include the power rule for radicals, which says \( \sqrt[n]{a^m} = a^{m/n} \). This can further be expanded upon by applying exponent laws like \( a^m \cdot a^n = a^{m+n} \) during simplification.

Such properties ensure the validity and correctness of the steps when breaking down or combining radical terms. By using them appropriately, we ensure that the algebraic manipulations we perform are both valid and efficient.

Simplifying Radicals

Simplifying radicals is a key objective when working with radical expressions. It means transforming an expression into its simplest form. This often involves breaking down composite numbers into their prime factors or recognizing perfect squares, cubes, etc.

For instance, to simplify \( \sqrt{x^7} \), you apply the property of radicals by rewriting it as \( (x^7)^{1/2} = x^{7/2} \). Similarly, \( \sqrt{y^8} \) becomes \( y^{8/2} = y^4 \). Using these exponent rules ensures the radical expressions are in their basic and simplified form.

To successfully simplify a radical expression:

• Look for any perfect squares or cubes that can be extracted from under the square root.
• Use exponent rules to break down the expression further.
• Combine any like terms or factors that can be simplified.

Simplifying radicals often involves recognizing patterns and practicing proper arithmetic with exponents to achieve the simplest result.

Product of Radicals

The concept of the product of radicals is fundamental in simplifying and calculating expressions that involve multiple radical terms. Utilizing the property \( \sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b} \) aids in breaking down seemingly complex radicals into more manageable pieces.

Consider the expression \( \sqrt{x^7 y^8} \). By applying the product of radicals property, you separate it into \( \sqrt{x^7} \cdot \sqrt{y^8} \). This involves confirming that both components, \( x^7 \) and \( y^8 \), are non-negative, which, in this exercise, they are stated to be because they represent non-zero real numbers.

Using this property effectively simplifies the expression while preserving its original value, making calculations easier to verify and understand. It demonstrates how expressions with shared components or factors can be split up and solved individually before being re-integrated, achieving more simplified solutions.