Problem 10 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ \left[10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4} g^{8}(2-k)^{17}\right]^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left[10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4} g^{8}(2-k)^{17}\right]^{4}$. Answer: The simplified expression is $10000t^{16}y^{28}j^{12}d^{8}v^{24}n^{16}g^{32}(2-k)^{68}$.

Step 1: Apply the power rule for powers

We have the given expression: $$ \left[10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4} g^{8}(2-k)^{17}\right]^{4} $$ Applying the power rule for powers, we get: $$ 10^4 \cdot t^{4\cdot4} \cdot y^{7\cdot4} \cdot j^{3\cdot4} \cdot d^{2\cdot4} \cdot v^{6\cdot4} \cdot n^{4\cdot4} \cdot g^{8\cdot4} \cdot (2-k)^{17\cdot4} $$

Step 2: Simplify the resulting expression

Now we need to simplify the resulting expression further by calculating the exponent values and the constant term. We get: $$ 10000 \cdot t^{16} \cdot y^{28} \cdot j^{12} \cdot d^{8} \cdot v^{24} \cdot n^{16} \cdot g^{32} \cdot (2-k)^{68} $$ So, the simplified expression is: $$ 10000t^{16}y^{28}j^{12}d^{8}v^{24}n^{16}g^{32}(2-k)^{68} $$

Key Concepts

Exponentiation

Exponentiation is an essential mathematical operation that involves raising numbers or expressions to a power. This power is represented as a small number written to the right and above the base number. For example, in the term \( t^4 \), \( t \) is the base, and \( 4 \) is the exponent, indicating that \( t \) is multiplied by itself four times.When dealing with exponentiation in algebra, some important rules help simplify expressions:

• Power Rule for Multiplication: When you multiply like bases raised to exponents, you add the exponents. For example, \( x^a \cdot x^b = x^{a+b} \).
• Power Rule for Powers: When raising a power to another power, multiply the exponents. For example, \( (x^a)^b = x^{a \, \times \, b} \).

In the given exercise, applying the power rule for powers helps us raise every term inside the brackets to the power of \( 4 \), which simplifies the expression efficiently.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the foundation of algebra and represent mathematical relationships using symbols.In the context of the exercise, we have an algebraic expression: \[10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4} g^{8}(2-k)^{17}\] Each element of this expression serves a specific purpose:

• Coefficients: These are the numbers in front of the variables, like \( 10 \) in \( 10 t^4 \), representing how many times the variables are multiplied.
• Variables: Letters such as \( t, y, j \), which stand as placeholders for numbers that may vary.
• Exponents: Small numbers indicating how many times the base variable is used as a factor, such as \(4\) in \(t^4\).

Understanding how to manipulate and simplify these expressions using algebraic rules is crucial for solving algebra problems effectively.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form without changing their value. It often requires applying the power rules and basic algebraic operations to combine like terms and eliminate unnecessary complexity.In the exercise, we simplified the expression:\[(10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4} g^{8}(2-k)^{17})^4\] Here's how simplification was achieved:

• Applied the power rule for powers, raising each term inside the parentheses to the fourth power.
• Calculated each variable's new exponent by multiplying the existing exponent by \(4\). For example, \( t^{4 \times 4} = t^{16} \).
• Multiplied the numeric coefficient \(10\) raised to the fourth power to get \( 10000 \).

Upon completing these steps, the expression was simplified to a more compact form:\[10000t^{16}y^{28}j^{12}d^{8}v^{24}n^{16}g^{32}(2-k)^{68}\]Simplifying expressions makes them easier to work with and is a critical skill in solving algebraic problems.