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Chapter 1: Problem 62

Simplify the given expression. $$ 5^{4}+2^{4} $$

Short Answer

Expert verified
The simplified expression is 641.

Step by step solution

01

Simplify the Powers

Evaluate each power separately. Calculate \(5^4\) and \(2^4\). \[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]\[ 2^4 = 2 \times 2 \times 2 \times 2 = 16 \]
02

Add the Results

Add the results from Step 1 to simplify the expression. \[ 625 + 16 = 641 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
In the realm of mathematics, exponents serve as a succinct way to denote repeated multiplication. An exponent tells you how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(5^4\), the base is 5 and the exponent is 4. This implies that you multiply 5 by itself four times: \(5 \times 5 \times 5 \times 5\). As a handy tip, remember that if any number is raised to the power of 1, it remains unchanged. Exponents are powerful tools that can make dealing with large numbers or repetitive multiplication much simpler and more streamlined. They also lay the groundwork for more advanced mathematical concepts.
Addition
Addition is one of the foundational operations in arithmetic. It involves combining or "summing" numbers to get a total. Consider the numbers 625 and 16, results from evaluating powers in our example. When you add these two results, \(625 + 16\), you get 641.

  • Start with the rightmost digits and add them together.
  • If the sum is 10 or more, carry over 1 to the next left-digit position.

The simplicity of addition makes it one of the first operations learned in math. Yet, it remains crucial in more complex problems and various fields like statistics, accounting, and engineering.
Simplification
Simplification in mathematics involves transforming a complex expression into a simpler or more manageable form. The aim is to make calculations and the interpretation of the results easier. For example, in our original expression, \(5^4 + 2^4\), we calculated each power separately to simplify the process: \(625\) and \(16\).

Here鈥檚 some simplification strategies:
  • Simplify expressions step-by-step, handling one operation at a time.
  • Use parentheses to guide the order of operations if needed.

Reduction of expressions to their simplest form not only aids in understanding but also in communicating mathematical ideas. It can also highlight underlying patterns or properties that might not be obvious otherwise.
Powers
Powers often go hand in hand with exponents. While exponents tell us how many times to multiply the base number, powers refer to the result of this multiplication. In our example, we simplified two powers: \(5^4\) which results in 625, and \(2^4\) which results in 16.

Some useful pointers about powers include:
  • A number raised to the power of 0 is always 1, except when the base is 0.
  • A negative base raised to an even exponent results in a positive number.
  • The power of 1 of any number is the number itself.

Understanding powers helps in grasping the growth of quantities, making them fundamental in fields such as physics, finance, and exponential growth models.

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Most popular questions from this chapter

Simplify the given expression. $$ 3^{3}-(7+6) $$

Simplify the given expression. $$ 2+9 \cdot[7+3 \cdot(9+5)] $$

Simplify the given expression. $$ 20 \cdot 17 \div 17 \cdot 14 $$

Simplify the given expression. $$ 5 \cdot 2+9 \cdot 8 $$

Simplify the given expression. $$ 15 \cdot 17+10 \div 10-12 \cdot 4 $$
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