Sure, let's solve the given equation step-by-step to find the value of [tex]\( n \)[/tex].
The given equation is:
[tex]\[ \frac{y^3 \times y^n}{y} = y^6 \][/tex]
1. Simplify the numerator:
Recall that when you multiply exponents with the same base, you add the exponents. So,
[tex]\[ y^3 \times y^n = y^{3+n} \][/tex]
2. Simplify the left-hand side further:
We now need to divide [tex]\( y^{3+n} \)[/tex] by [tex]\( y \)[/tex]. When you divide exponents with the same base, you subtract the exponents. Therefore,
[tex]\[ \frac{y^{3+n}}{y} = y^{3+n-1} \][/tex]
So, the equation simplifies to:
[tex]\[ y^{3+n-1} = y^6 \][/tex]
3. Set the exponents equal to each other:
Since the bases [tex]\( y \)[/tex] on both sides of the equation are the same, we can equate the exponents:
[tex]\[ 3 + n - 1 = 6 \][/tex]
4. Solve for [tex]\( n \)[/tex]:
Simplify the exponent equation:
[tex]\[ 3 + n - 1 = 6 \][/tex]
[tex]\[ n + 2 = 6 \][/tex]
Subtract 2 from both sides to isolate [tex]\( n \)[/tex]:
[tex]\[ n = 6 - 2 \][/tex]
[tex]\[ n = 4 \][/tex]
So, the value of [tex]\( n \)[/tex] is [tex]\( 4 \)[/tex].
