Problem 71 - Algebra 1

Question

Simplify the expression. Then use a calculator to evaluate the expression. Round the result to the nearest tenth when appropriate. $$ 2.6^{4} \cdot 2.6^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(2.6^6\). After calculation and rounding, the final result is 604.66.

Step 1: Applying Exponential Law

The exponential multiplication law states that \( a^m \cdot a^n = a^{m+n} \). This law is applicable as we have the same base multiplied but with different exponents. Therefore, the expression could be simplified as \( 2.6^4 \cdot 2.6^2 = 2.6^{4+2} = 2.6^6 \)

Step 2: Computing Expression

Next, a calculator is used to compute the value of the expression \(2.6^6\). Remember to round the result to the nearest tenth for the final result.

Step 3: Rounding Off

The solution from the calculator needs to be rounded to the nearest tenth. The rounding process involves the examination of the tenth decimal place. If it's 5 or greater, the number in the ones place is increased by one. If it is less than 5, the number in the ones place remains the same.

Key Concepts

Exponential Laws

The laws of exponents are critical for simplifying exponential expressions. These mathematical rules dictate how to handle operations involving exponents. One of the fundamental exponential laws is the product of powers law, which states that when multiplying two exponents with the same base, you can add the exponents together. The general form of this law is \(a^m \cdot a^n = a^{m+n}\). In practice, if you have \(2^3 \cdot 2^4\), it simplifies to \(2^{3+4} = 2^7\), because both exponents have the same base of 2. By understanding and applying exponential laws effectively, complicated expressions become much simpler to manage and calculate.

Rounding Decimals

Rounding decimals helps to simplify numerical values to a specific precision, often making them easier to work with or communicate. The process involves looking at the decimal digit immediately after the place value to which you are rounding. For instance, if you are rounding to the nearest tenth, look at the hundredths place. If this digit is 5 or more, increase the tenths digit by one. If it's less than 5, leave the tenths digit as it is. It's crucial, especially in scientific and financial calculations, where a high level of precision is not always necessary, allowing for results that are quicker to read and understand.

Calculating Exponents

When calculating exponents, it's important to understand that the exponent denotes how many times the base is multiplied by itself. For example, \(3^4\) means \(3 \cdot 3 \cdot 3 \cdot 3\). To calculate exponents without a calculator, start by multiplying the base by itself repeatedly until you've reached the number of times indicated by the exponent. With a calculator, you simply input the base, use the exponentiation function (often labeled as '^' or 'exp'), and enter the exponent. Calculating exponents quickly and accurately is essential in many fields, including physics, engineering, and economics.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a particular value or set of values. In the realm of algebra, expressions can be simplified using various properties and laws, like distributive, associative, and commutative properties, alongside exponential laws. Simplifying an expression can help with understanding the relationships between variables and numbers, and aid in solving equations more efficiently. Mathematicians and students alike use these expressions as a fundamental part of problem solving and logical reasoning.