Problem 49 - Elementary Algebra

Question

Simplify \(\left(4 x^{5} y^{2}\right)^{3}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \((4x^5y^2)^3\). Answer: The simplified expression is \(64x^{15}y^6\).

Step 1: Distribute the exponent to each factor

Using the second exponent property, distribute the exponent 3 to each factor inside the parentheses: \((4x^5y^2)^3 = 4^3 x^{5\cdot3} y^{2\cdot3}\).

Step 2: Apply the exponent property#a^m^n = a^{mn}

Now apply the first exponent property to find the new exponents for \(x\) and \(y\): \(x^{5\cdot3} = x^{15}\), and \(y^{2\cdot3} = y^6\).

Step 3: Compute \(4^3\)

Compute \(4^3 = 4\times4\times4=64\).

Step 4: Combine the simplified factors

Finally, combine all the simplified factors to obtain the final expression: \(64x^{15}y^6\). So, the simplified expression is \(\boxed{64x^{15}y^6}\).

Key Concepts

Exponent Properties

Understanding the properties of exponents is crucial when working with algebraic expressions involving exponential notation. When simplifying expressions like \(\left(4 x^{5} y^{2}\right)^{3}\), we apply specific exponent rules to make the process systematic and straightforward. The most commonly used properties include the Product of Powers rule, which states that when multiplying two powers with the same base, you add the exponents \(x^a \cdot x^b = x^{a+b}\). Another property is the Power of a Power rule, which informs us to multiply the exponents when a power is raised to another power \(\left(x^a\right)^b = x^{a\cdot b}\).

For example, in the given exercise, the expression inside the parentheses is raised to the third power, prompting the use of the Power of a Power rule. By applying this rule, each base \(x\) and \(y\) gets its exponent multiplied by 3, transforming \(x^5\) into \(x^{15}\) and \(y^2\) into \(y^6\). This property simplifies the expression significantly, enabling more straightforward multiplication and combination of terms.

Algebraic Manipulation

Algebraic manipulation refers to the various methods used to rewrite and simplify algebraic expressions. It involves a set of techniques that allow us to move terms, factors, and variables around in an equation or expression while keeping their values equivalent. In the context of simplifying \(\left(4 x^{5} y^{2}\right)^{3}\), algebraic manipulation comes into play when we distribute the exponent across the terms inside the parentheses.

Initially, the expression might seem complex, but by applying exponent properties step by step, the manipulation process becomes clear. By distributing the power of 3 to each base - the coefficient 4, variable \(x\), and variable \(y\) - and then operating on them separately, every term becomes simpler to handle. Afterwards, combining these terms as shown in the final step of the solution yields a much simpler expression that maintains its original value. These techniques form the basis for increasing complexity in algebra and are essential for higher levels of mathematics.

Exponential Notation

Exponential notation is a concise way to represent repeated multiplication of the same factor. In the notation \(a^n\), \(a\) is the base and \(n\) is the exponent, indicating that \(a\) is to be multiplied by itself \(n\) times. In the exercise \(\left(4 x^{5} y^{2}\right)^{3}\), \(4\), \(x\), and \(y\) are the bases, and the numbers 5, 2, and 3 are exponents.

The convenience of exponential notation becomes clear when dealing with large numbers or many repetitions of the same base. Instead of writing out \(4 \cdot 4 \cdot 4\), it is more efficient to use \(4^3\), especially when the numbers get even larger. Furthermore, the capability of exponential notation to condense extensive multiplication processes into compact terms makes algebraic manipulation and simplification far more manageable. The essence of understanding exponential notation lies in recognizing how much simpler it makes reading, writing, and calculating expressions in mathematics.