Question

Как умножить степени с разными основаниями и показателями?

Answer 1

1) Если умножаются 2 числа с одинаковыми основаниями, но разными показателями, то общее основание возводится в сумму степеней.:

$a^n*a^m=a^{n+m}$

Пример
3⁴*3³=3⁴⁺³=3⁷

2) Если основания разные, а показатели одинаковые. В этом случае мы возводим в степень произведение оснований.
aⁿ*bⁿ=(ab)ⁿ

Пример:
5²*2²=(5*2)²=10²=100
3) Если основания разные и показатели разные, то тут 2 варианта:
1. Выделяем одинаковое основание, т.е. раскладываем один из множителей.

Представим число b=a*c
$a^n*b^m$$a^n*b^m=a^n*(ac)^m=a^{n+m}*c^m$

Пример
$2^3*14^2=2^3*(7*2)^2=2^{2+3}*7^2=2^5*7^2=32*49=1568$

2. Приводим к общему показателю:
$a^n*b^m=a^n*b^{n}*b^{m-n}=(ab)^n*b^{m-n}$

Пример
$3^4*2^5=3^4*2^4*2=6^4*2=2592$

Answer 2

Multiplying Powers with Different Bases and Exponents

When multiplying powers with different bases and exponents, you can use the following rules to simplify the expression:

1. Multiplying Powers with the Same Base: - When multiplying powers with the same base, you can add the exponents. - For example, a^m * a^n = a^(m+n).

**Source* provides an example: "a^3 * b^2 = a^3 + b^2 = a^3b^2."

2. Multiplying Powers with Different Bases: - When multiplying powers with different bases, you cannot directly simplify the expression unless the bases are related in some way (e.g., one base is a power of the other). - For example, a^m * b^n cannot be simplified further unless there is a specific relationship between a and b.

3. Multiplying Powers with Different Bases and Exponents: - If you have powers with different bases and exponents, you cannot simplify the expression further without additional information about the relationship between the bases.

4. Using Logarithms: - If you need to calculate the value of an expression involving powers with different bases and exponents, you can use logarithms to simplify the calculation. - For example, if you need to calculate 2^3 * 3^2, you can use logarithms to simplify the expression.

**Source* provides an example: "log2(4) = 2. This means 2^3 = 8. Similarly, log3(9) = 2. This means 3^2 = 9. Therefore, 2^3 * 3^2 = 8 * 9 = 72."

In summary, when multiplying powers with different bases and exponents, you can simplify the expression if the bases are the same by adding the exponents. If the bases are different, the expression cannot be further simplified without additional information about the relationship between the bases. If you need to calculate the value of such an expression, you can use logarithms to simplify the calculation.