Question

If \(3^{3} \times 9^{12}=3^{\mathrm{x}},\) what is the value of \(x ?\)

Short answer

In conclusion, the value of \(x\) is 27.

Step-by-step solution

Express 9 as a Power of 3

Since 9 is the square of 3, it can be expressed as \(3^2\). Therefore, we can rewrite \(9^{12}\) as \((3^2)^{12}\) in the given equation.

Use properties of Exponents

The law of exponentiation tells us that \((a^m)^n=a^{mn}\). We apply this rule to \((3^2)^{12}\) which is equal to \(3^{2 \times 12}\), and our equation becomes \(3^3 \times 3^{24} = 3^x\).

Combine like Bases

When multiplying exponents with the same base, we can simply add the exponent values together. This gives us \(3^{(3+24)}=3^x\). In other words, \(3^{27} = 3^x\).

Compare Exponent on Both Sides

Now that both sides of the equation have the same base, \(x\) must be equal to the power on the left hand side. Hence our solution is \(x=27\).

Key concepts

Exponents

Exponents are a fundamental part of mathematics and are used to express repeated multiplication of a number by itself. An exponent is written as a small number called the "power" placed above and to the right of the base number. For example, in the expression \(3^4\), 3 is the base and 4 is the exponent, indicating that 3 is multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3\).

Exponents follow specific rules that make calculations easier. Some of these include:

• **Product of Powers Rule:** When multiplying two exponents with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Power of a Power Rule:** When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• **Power of a Product Rule:** Apply the exponent to each factor inside the parentheses: \((ab)^m = a^m \times b^m\).

Recognizing and applying these rules is crucial when solving any GRE Math Problems involving exponents. Understanding these basics can simplify complex algebraic equations and aid in solving mathematical problems more efficiently.

Mathematical Properties

Mathematical properties are the foundational rules that apply to various operations and expressions in mathematics. These properties ensure that mathematical calculations are consistent and reliable, which is especially important in the field of algebra.

For the exponent problem we considered, several key mathematical properties come into play:

• **Commutative Property of Multiplication:** This property states that changing the order of factors does not change the product: \(a \times b = b \times a\).
• **Associative Property of Multiplication:** This property indicates that the way in which factors are grouped does not affect their product: \((a \times b) \times c = a \times (b \times c)\).
• **Distributive Property:** Although not directly used in this problem, knowing that you can distribute a multiplied value across terms in parentheses is helpful: \(a(b+c) = ab + ac\).

These properties are crucial for simplifying expressions and solving equations, and they streamline the process of handling complex problems. Understanding their application can greatly aid in solving GRE Math Problems involving algebraic equations.

Algebraic Equations

Algebraic equations form the basis of many mathematical problems, especially those seen on exams like the GRE. They consist of variables and constants, and the goal is often to solve for a variable by performing a series of calculations that maintain the balance of the equation.

In the context of the given exercise, the algebraic equation \(3^3 \times 9^{12}=3^{x}\) was simplified using properties of exponents to find \(x\). Here’s a simplified explanation of the approach:

• **Expression Simplification:** Begin by expressing 9 as a power of 3, so \(9^{12}\) becomes \((3^2)^{12}\).
• **Utilizing Exponent Rules:** Use the power of a power rule to transform \((3^2)^{12}\) into \(3^{24}\).
• **Combining Like Bases:** With a common base of 3, add the exponents as per the product of powers rule to solve \(3^3 \times 3^{24} = 3^{27}\), confirming \(x=27\).

This systematic approach using algebraic equations and their properties allows for effective problem-solving and is a critical skill for GRE Math problems. Understanding the balance and manipulation of equations forms the foundation for tackling a wide array of mathematical challenges.