Question

Simplify. $$ (-3 x y)^{3}(2 y)^{2} $$

Short answer

The simplified form of the given expression is \( -108x^{3}y^{5} \)

Step-by-step solution

Identify the Base and Power

The given problem is \( (-3 x y)^{3}(2 y)^{2} \). Here, -3, x and y are the first set of base numbers with the power being 3 and 2 and y are the second set of base numbers with power being 2.

Apply the Rule

Now apply the rules of indices which states that \( (ab)^n = a^n b^n \) to both sets. For the first set (-3 x y)3 we have \( (-3)^{3} (x)^{3} (y)^{3} \), and for the second set (2 y)2 we have \( (2)^{2} (y)^{2} \)

Simplify the Powers

Simplify the expression by calculating the powers: \( (-3)^{3} = -27 \), \( (2)^{2} = 4 \). Substituting these into the expression, we obtain \( -27x^{3}y^{3} \times 4y^{2} \)

Simplify the Expression

Simplify the expression by multiplying the constants -27 and 4 to get -108, and applying the rule for multiplying terms with the same base (i.e. \(a^n \times a^m = a^{n+m}\)) to get \(y^{3+2}\) . Therefore, the simplified expression is \( -108x^{3}y^{5} \)

Key concepts

Rules of Indices

Understanding the rules of indices, also known as the laws of exponents, is crucial for simplifying algebraic expressions involving powers. These rules tell us how to handle exponents when multiplying, dividing, raising powers to another power, and more. For instance, one important rule is that when you multiply two terms with the same base, you add the exponents: \(a^n \times a^m = a^{n+m}\). Another rule applies when raising a power to a power, where you multiply the exponents: \( (a^n)^m = a^{n \times m}\). These rules streamline the process of working with powers and make complex algebraic expressions more manageable.

Exponents and Powers

Exponents and powers represent the number of times a base number is multiplied by itself. An expression like \(a^n\) indicates that the base \(a\) is to be multiplied by itself \(n\) times. The exponent \(n\) can be any integer—positive, negative, or zero. When we encounter negative exponents, such as \(a^{-n}\), this represents \(1 / a^n\), or the reciprocal of the base raised to a positive power. Zero as an exponent, as in \(a^0\), always equals one, since any number to the power of zero is one. Understanding these principles helps us simplify expressions efficiently.

Multiplying Algebraic Terms

When multiplying algebraic terms, we must consider both the coefficients (numerical parts) and the variables. The key to multiplying terms correctly involves multiplying the coefficients together and applying the rules of indices for the variables. For expressions with the same variable base, we add the exponents as previously mentioned. So \( 3x^2 \times 2x^3 = 6x^{2+3} = 6x^5\). It's essential to handle each part of the term—coefficient and variable—separately and then combine them for the final simplified form.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division and work in conjunction with the distributive, associative, and commutative properties. These operations allow us to manipulate and simplify expressions. Multiplication of expressions, for example, often involves the distributive property, where \(a(b + c) = ab + ac\). When we simplify algebraic expressions, we typically aim to combine like terms—those terms with the same variables and exponents—and perform the operations according to the precedence. As we follow these operational rules properly, algebraic expressions become much easier to solve.