We consider the testing hypothesis that two random vectors of p and q components are independent in canonical correlation analysis. In this paper we investigate the powers of the test based on the largest root criterion. As the exact distribution are expressed by the zonal polynomials, the computation is possible only for p=2, and also it is necessary to calculate using quadruplex precision because we lose the significance by subtraction. So in Table I we obtain the percentage points of the largest root criterion for the computation of the quadruplex precision. Then we calculate the power when p=2 and q = 3 to 11 (2). The results show that for the fixed n–q the power becomes smaller when q increases, and for the fixed p1 of the alternative hypothesis (p1, P2) the power does not become significantly large when P2 increases. We can also find the sample size required for the power agnist some alternative hypothesis to be about 0.9. the numerical results may be useful to find the quality of approximation by using formula of the asyptotic distribution.