Solve 2a² × 3a⁴: Multiplying Algebraic Expressions with Powers

Given that there is a multiplication between all terms in the expression, we will apply the distributive property of multiplication. This allows us to handle the coefficients of terms raised to powers, as well as the terms themselves separately. For added clarity, let's handle it in steps:

$2a^2\cdot3a^4=2\cdot3\cdot a^2\cdot a^4=6\cdot a^2\cdot a^4$

Due to the multiplication between all terms we could do this, it should be noted that we can (and it's preferable to) skip the middle step, meaning:

Write directly:$2a^2\cdot3a^4=6\cdot a^2\cdot a^4$

From here on we will no longer write the multiplication sign. We will instead place the terms next to each other\ place the term next to its coefficient to indicate multiplication between them,

Proceed to apply the law of exponents for multiplication of terms with identical bases:

$c^m\cdot c^n=c^{m+n}$

Note that this law applies to any number of terms being multiplied and not just two, for example for three terms with identical bases we obtain:

$c^m\cdot c^n\cdot c^k=c^{m+n}\cdot c^k=c^{m+n+k}$

Whilst we used the law of exponents for two terms we can equally perform the same calculation for four terms or five and so on..,

Let's return to the problem and apply the above law of exponents:

$6a^2a^4=6a^{2+4}=6a^6$

Therefore the correct answer is C.

Important note:

Here we need to emphasize that we should always ask the question - what is the exponent being applied to?

For example, in this problem the exponent applies only to the bases of

$a$$$and not to the numbers, more clearly, in the following expression: $5c^7$the exponent applies only to $c$ and not to the number 5,

whereas when we write:$(5c)^7$the exponent applies to each term of the multiplication inside the parentheses,

meaning:$(5c)^7=5^7c^7$This is actually the application of the law of exponents:

$(x\cdot y)^n=x^n\cdot y^n$

which follows both from the meaning of parentheses and from the definition of exponents.