Problem 69 - Algebra 1

Question

Copy the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Power } & 9^{1} & 9^{2} & 9^{3} & 9^{4} & 9^{5} & 9^{6} & 9^{7} & 9^{8} \\ \hline \text { Evaluate } & ? & ? & ? & ? & ? & ? & ? & ? \\ \hline \end{array} $$ Evaluate the powers of 9 in the table. What pattern do you see for the last digit of each product?

Step-by-Step Solution

Answer

The evaluation of the powers of 9 from \(9^{1}\) to \(9^{8}\) yields 9, 81, 729, 6561, 59049, 531441, 4782969, and 43046721 respectively. The pattern in the last digit for each product is that it alternates between 1 and 9.

Step 1: Calculation of the powers of 9

Calculate the value of each given power:\n- \(9^{1} = 9\),- \(9^{2} = 81\),- \(9^{3} = 729\),- \(9^{4} = 6561\),- \(9^{5} = 59049\),- \(9^{6} = 531441\),- \(9^{7} = 4782969\),- \(9^{8} = 43046721\).

Step 2: Observing the last digit in each product

The last digit in each of the products are as follows:- \(9^{1} = 9\),- \(9^{2} = 81\),- \(9^{3} = 729\),- \(9^{4} = 6561\),- \(9^{5} = 59049\),- \(9^{6} = 531441\),- \(9^{7} = 4782969\),- \(9^{8} = 43046721\).Thus, for the power of 1, the last digit is 9. For powers of 2 and onwards, the digit in the ones place follows a pattern of alternating between 1 and 9.

Key Concepts

Patterns in mathematics

Patterns are fundamental in mathematics. They help simplify complex problems and reveal underlying structures. In the exercise provided, we observe a numerical pattern by looking at the powers of 9. Here's what this means.
When you first calculate powers, like in the table from the exercise, they might just look like a series of numbers. However, by focusing on particular aspects, such as the last digit, intriguing patterns emerge. In this case, powers of 9 exhibit a cyclic pattern.

• The last digit of odd powers of 9 is always 9.
• The last digit of even powers of 9 always ends in 1.

This cyclical change repeats itself as you calculate further powers of 9. Recognizing these can aid in predicting and verifying results without full calculations, emphasizing the power of patterns in mathematical reasoning.

Power rules

Power rules are a vital component of exponent operations, which govern how we handle numbers involved in power expressions. Commonly, a power expression appears as a base number raised to an exponent, written as \(a^n\).
Some fundamental power rules include:

• Product of powers rule: When multiplying two powers with the same base, you add their exponents, \(a^m \times a^n = a^{m+n}\).
• Quotient of powers rule: When dividing two powers with the same base, subtract their exponents, \(a^m / a^n = a^{m-n}\).
• Power of a power rule: When raising a power to another power, you multiply the exponents, \((a^m)^n = a^{m \times n}\).

These rules provide a foundation to manipulate and simplify expressions involving exponents efficiently. By understanding and applying these rules, complex calculations can often be reduced to more manageable numbers, much like how we dealt with powers of 9 in the exercise.

Multiplication of numbers based on exponents

Multiplication involving exponents involves applying the power rules to simplify the process. In our exercise with powers of 9, we see these rules in action.
When multiplying powers that share the same base, such as \(9^2 \times 9^3\), you simplify the operation by adding the exponents, using the product of powers rule. This converts the expression into \(9^{2+3} = 9^5\).
Thus, the process relies on:

• Identifying common bases in multiplicative expressions.
• Applying power rules to manage computations and reduce expression complexity.
• Using patterns and last-digit observations to check for accuracy, which helps in faster calculations.

Understanding these concepts not only aids in the specific task of multiplying powers but also builds invaluable skills applicable in various arithmetic and algebraic contexts.