Which expression is equivalent to 102 · 104?
A) 102
B) 104
C) 106
D) 1006
The Correct answer and Explanation is:
The given expression is 102⋅10410^2 \cdot 10^4102⋅104. To simplify this expression, you can apply the property of exponents, which states:am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n
In this case, a=10a = 10a=10, m=2m = 2m=2, and n=4n = 4n=4. Using the exponent rule:102⋅104=102+4=10610^2 \cdot 10^4 = 10^{2+4} = 10^6102⋅104=102+4=106
Thus, the expression 102⋅10410^2 \cdot 10^4102⋅104 is equivalent to 10610^6106, which corresponds to option C.
Explanation:
Exponent rules are essential tools in algebra that simplify expressions involving powers of the same base. The base in the given problem is 10, and it appears in both terms of the expression, with the powers 2 and 4 respectively. When multiplying terms with the same base, we add the exponents. This rule stems from the definition of exponents as repeated multiplication.
Let’s break this down:
- 10210^2102 means 10×1010 \times 1010×10, which equals 100.
- 10410^4104 means 10×10×10×1010 \times 10 \times 10 \times 1010×10×10×10, which equals 10,000.
When multiplying these two numbers:100×10,000=1,000,000=106100 \times 10,000 = 1,000,000 = 10^6100×10,000=1,000,000=106
Thus, 102⋅10410^2 \cdot 10^4102⋅104 simplifies to 10610^6106, following the exponent addition rule. Understanding this property helps avoid unnecessary calculations and leads directly to the correct answer.
The other answer options are incorrect:
- Option A (10210^2102) and Option B (10410^4104) are smaller powers of 10, and Option D (10100610^{1006}101006) is far too large and irrelevant to the given problem. Therefore, the correct answer is C) 10610^6106.